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Linear map

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</style><div role="note" class="hatnote navigation-not-searchable">"Linear transformation" redirects here. For fractional linear transformations, see <a href="M%C3%B6bius_transformation" title="Möbius transformation">Möbius transformation</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">"Linear Operators" redirects here. For the textbook by Dunford and Schwarz, see <a href="Linear_Operators_(book)" title="Linear Operators (book)">Linear Operators (book)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Linear_function" title="Linear function">linear function</a>.</div>

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In <a href="Mathematics" title="Mathematics">mathematics</a>, and more specifically in <a href="Linear_algebra" title="Linear algebra">linear algebra</a>, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a <a href="Map_(mathematics)" title="Map (mathematics)">mapping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\to W}">

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<annotation encoding="application/x-tex">{\displaystyle V\to W}</annotation>

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</math><img src="./98b0589697c495103fca82279cf0c84b944b62b6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.837ex; height:2.176ex;" alt="{\displaystyle V\to W}" loading="lazy"> between two <a href="Vector_space" title="Vector space">vector spaces</a> that preserves the operations of <a href="Vector_addition" class="mw-redirect" title="Vector addition">vector addition</a> and <a href="Scalar_multiplication" title="Scalar multiplication">scalar multiplication</a>. The same names and the same definition are also used for the more general case of <a href="Module_(mathematics)" title="Module (mathematics)">modules</a> over a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a>; see <a href="Module_homomorphism" title="Module homomorphism">Module homomorphism</a>.

If a linear map is a <a href="Bijection" title="Bijection">bijection</a> then it is called a <style data-mw-deduplicate="TemplateStyles:r1238216509">

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</style>linear isomorphism. In the case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=W}">

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<annotation encoding="application/x-tex">{\displaystyle V=W}</annotation>

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</math><img src="./740038d36bd79466d6938d73b83fe737161fa1c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.321ex; height:2.176ex;" alt="{\displaystyle V=W}" loading="lazy">, a linear map is called a linear endomorphism. Sometimes the term linear operator refers to this case,<a href="#cite_note-1">[1]</a> but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> are <a href="Real_number" title="Real number">real</a> vector spaces (not necessarily with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V=W}">

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<annotation encoding="application/x-tex">{\displaystyle V=W}</annotation>

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</math><img src="./740038d36bd79466d6938d73b83fe737161fa1c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.321ex; height:2.176ex;" alt="{\displaystyle V=W}" loading="lazy">), or it can be used to emphasize that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> is a <a href="Function_space" title="Function space">function space</a>, which is a common convention in <a href="Functional_analysis" title="Functional analysis">functional analysis</a>.<a href="#cite_note-2">[2]</a> Sometimes the term <a href="Linear_function" title="Linear function">linear function</a> has the same meaning as linear map, while in <a href="Mathematical_analysis" title="Mathematical analysis">analysis</a> it does not.

A linear map from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> always maps the origin of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> to the origin of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy">. Moreover, it maps <a href="Linear_subspace" title="Linear subspace">linear subspaces</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> onto linear subspaces in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> (possibly of a lower <a href="Dimension_(vector_space)" title="Dimension (vector space)">dimension</a>);<a href="#cite_note-3">[3]</a> for example, it maps a <a href="Plane_(geometry)" class="mw-redirect" title="Plane (geometry)">plane</a> through the <a href="Origin_(geometry)" class="mw-redirect" title="Origin (geometry)">origin</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> to either a plane through the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy">, a <a href="Line_(geometry)" title="Line (geometry)">line</a> through the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy">, or just the origin in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy">. Linear maps can often be represented as <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>, and simple examples include <a href="Rotations_and_reflections_in_two_dimensions" title="Rotations and reflections in two dimensions">rotation and reflection linear transformations</a>.

In the language of <a href="Category_theory" title="Category theory">category theory</a>, linear maps are the <a href="Morphism" title="Morphism">morphisms</a> of vector spaces, and they form a category <a href="Equivalence_of_categories" title="Equivalence of categories">equivalent</a> to <a href="Category_of_matrices" title="Category of matrices">the one of matrices</a>.

<div class="mw-heading mw-heading2"><h2 id="Definition_and_first_consequences">Definition and first consequences</h2></div>

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> be vector spaces over the same <a href="Field_(mathematics)" title="Field (mathematics)">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

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<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

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</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy">.

A <a href="Function_(mathematics)" title="Function (mathematics)">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:V\to W}">

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<annotation encoding="application/x-tex">{\displaystyle f:V\to W}</annotation>

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</math><img src="./574dffa1c85efaef6b6ef553ebd8ad9cf7f87fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\displaystyle f:V\to W}" loading="lazy"> is said to be a linear map if for any two vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {u} ,\mathbf {v} \in V}">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {u} ,\mathbf {v} \in V}</annotation>

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</math><img src="./f683aa60560de6fabc20a35c25e37d485cd6c1ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.558ex; height:2.509ex;" alt="{\textstyle \mathbf {u} ,\mathbf {v} \in V}" loading="lazy"> and any scalar <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c\in K}">

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<annotation encoding="application/x-tex">{\displaystyle c\in K}</annotation>

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</math><img src="./a5e894da36b4db8c32e35962d3950750ad556f46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.913ex; height:2.176ex;" alt="{\displaystyle c\in K}" loading="lazy"> the following two conditions are satisfied:

<ul><li><a href="Additive_map" title="Additive map">Additivity</a> / operation of addition <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {u} +\mathbf {v} )=f(\mathbf {u} )+f(\mathbf {v} )}">

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<annotation encoding="application/x-tex">{\displaystyle f(\mathbf {u} +\mathbf {v} )=f(\mathbf {u} )+f(\mathbf {v} )}</annotation>

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</math></li>

<li><a href="Homogeneous_function" title="Homogeneous function">Homogeneity</a> of degree 1 / operation of scalar multiplication <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(c\mathbf {u} )=cf(\mathbf {u} )}">

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<annotation encoding="application/x-tex">{\displaystyle f(c\mathbf {u} )=cf(\mathbf {u} )}</annotation>

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</math></li></ul>

Thus, a linear map is said to be operation preserving. In other words, it does not matter whether the linear map is applied before (the right hand sides of the above examples) or after (the left hand sides of the examples) the operations of addition and scalar multiplication.

By <a href="Addition#Associativity" title="Addition">the associativity of the addition operation</a> denoted as +, for any vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}\in V}">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}\in V}</annotation>

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</math><img src="./e4ff56d3267c11c192954e9af7dff69901de29dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.049ex; height:2.509ex;" alt="{\textstyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}\in V}" loading="lazy"> and scalars <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle c_{1},\ldots ,c_{n}\in K,}">

<msub>

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<annotation encoding="application/x-tex">{\textstyle c_{1},\ldots ,c_{n}\in K,}</annotation>

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</math><img src="./c74cdae5558856a84f5e91fd0c9f19af18a785da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.018ex; height:2.509ex;" alt="{\textstyle c_{1},\ldots ,c_{n}\in K,}" loading="lazy"> the following equality holds:<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle f(c_{1}\mathbf {u} _{1}+\cdots +c_{n}\mathbf {u} _{n})=c_{1}f(\mathbf {u} _{1})+\cdots +c_{n}f(\mathbf {u} _{n}).}</annotation>

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</math> Thus a linear map is one which preserves <a href="Linear_combination" title="Linear combination">linear combinations</a>.

Denoting the zero elements of the vector spaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

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<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

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</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

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<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

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</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {0} _{V}}">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {0} _{V}}</annotation>

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</math><img src="./6dae4bf2e588270286c59735162368e5ee0cc6af.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.833ex; height:2.509ex;" alt="{\textstyle \mathbf {0} _{V}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {0} _{W}}">

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<annotation encoding="application/x-tex">{\textstyle \mathbf {0} _{W}}</annotation>

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</math><img src="./92ec046704f3874e6e8aadcb54c8004ef81ced79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.291ex; height:2.509ex;" alt="{\textstyle \mathbf {0} _{W}}" loading="lazy"> respectively, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(\mathbf {0} _{V})=\mathbf {0} _{W}.}">

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</mrow>

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<annotation encoding="application/x-tex">{\textstyle f(\mathbf {0} _{V})=\mathbf {0} _{W}.}</annotation>

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</math><img src="./8cfb881efeac08d176d7cb5909e700529465bb5b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.957ex; height:2.843ex;" alt="{\textstyle f(\mathbf {0} _{V})=\mathbf {0} _{W}.}" loading="lazy"> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c=0}</annotation>

</semantics>

</math><img src="./d9ee918699d0cb4b8c633cc1f520a8a7a174f44a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.268ex; height:2.176ex;" alt="{\displaystyle c=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {v} \in V}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {v} \in V}</annotation>

</semantics>

</math><img src="./557f9cbf990da31ddc7aa25ce3a7be5f87ee04d4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.039ex; height:2.176ex;" alt="{\textstyle \mathbf {v} \in V}" loading="lazy"> in the equation for homogeneity of degree 1:

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\mathbf {0} _{V})=f(0\mathbf {v} )=0f(\mathbf {v} )=\mathbf {0} _{W}.}</annotation>

</semantics>

</math>

A linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\to K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V\to K}</annotation>

</semantics>

</math><img src="./14f40344b7f468c6d9e2eae03f37ee6286f19d3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.467ex; height:2.176ex;" alt="{\displaystyle V\to K}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K}</annotation>

</semantics>

</math><img src="./2b76fce82a62ed5461908f0dc8f037de4e3686b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" loading="lazy"> viewed as a one-dimensional vector space over itself is called a <a href="Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a>.<a href="#cite_note-6">[6]</a>

These statements generalize to any left-module <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {}_{R}M}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {}_{R}M}</annotation>

</semantics>

</math><img src="./0b8868607ccc60aefecc3f650b7466a8ae6c67a1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.922ex; height:2.509ex;" alt="{\textstyle {}_{R}M}" loading="lazy"> over a ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> without modification, and to any right-module upon reversing of the scalar multiplication.

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<ul><li>A prototypical example that gives linear maps their name is a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:\mathbb {R} \to \mathbb {R} :x\mapsto cx}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:\mathbb {R} \to \mathbb {R} :x\mapsto cx}</annotation>

</semantics>

</math><img src="./1f558ca3675215137a38ff8742a10eebb7e3034e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.403ex; height:2.509ex;" alt="{\displaystyle f:\mathbb {R} \to \mathbb {R} :x\mapsto cx}" loading="lazy">, of which the <a href="Graph_of_a_function" title="Graph of a function">graph</a> is a line through the origin.<a href="#cite_note-7">[7]</a></li>

<li>More generally, any <a href="Homothety" title="Homothety">homothety</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {v} \mapsto c\mathbf {v} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {v} \mapsto c\mathbf {v} }</annotation>

</semantics>

</math><img src="./18bcdb55dbdd5c5d2b60de7f21f00771396b1d81.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.443ex; height:1.843ex;" alt="{\textstyle \mathbf {v} \mapsto c\mathbf {v} }" loading="lazy"> centered in the origin of a vector space is a linear map (here c is a scalar).</li>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {x} \mapsto \mathbf {0} }</annotation>

</semantics>

</math><img src="./812b31dd59a14f41d4178bb777317bd3bdd88f37.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.362ex; height:2.176ex;" alt="{\textstyle \mathbf {x} \mapsto \mathbf {0} }" loading="lazy"> between two vector spaces (over the same <a href="Field_(mathematics)" title="Field (mathematics)">field</a>) is linear.</li>

<li>The <a href="Identity_function" title="Identity function">identity map</a> on any module is a linear operator.</li>

<li>For real numbers, the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x\mapsto x^{2}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x\mapsto x^{2}}</annotation>

</semantics>

</math><img src="./b0bcf676f15aa22ab2429aeb7935d84835df8a9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.328ex; height:2.509ex;" alt="{\textstyle x\mapsto x^{2}}" loading="lazy"> is not linear.</li>

<li>For real numbers, the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x\mapsto x+1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x\mapsto x+1}</annotation>

</semantics>

</math><img src="./a43c220f372775e932f4d93e7d374279e91bda7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.276ex; height:2.343ex;" alt="{\textstyle x\mapsto x+1}" loading="lazy"> is not linear (but is an <a href="Affine_transformation" title="Affine transformation">affine transformation</a>).</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation>

</semantics>

</math><img src="./12b23d207d23dd430b93320539abbb0bde84870d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}" loading="lazy"> <a href="Real_matrix" class="mw-redirect" title="Real matrix">real matrix</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> defines a linear map from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./c510b63578322050121fe966f2e5770bea43308d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{m}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{m}}</annotation>

</semantics>

</math><img src="./6a87a024931038d1858dc22e8a194e5978c3412e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.353ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{m}}" loading="lazy"> by sending a <a href="Column_vector" class="mw-redirect" title="Column vector">column vector</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}</annotation>

</semantics>

</math><img src="./f33c7feddfbe1e5c87647a55639651d9fb2f23de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.148ex; height:2.343ex;" alt="{\displaystyle \mathbf {x} \in \mathbb {R} ^{n}}" loading="lazy"> to the column vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {x} \in \mathbb {R} ^{m}}">

</mrow>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} \in \mathbb {R} ^{m}}</annotation>

</semantics>

</math><img src="./ff693d6fcdb81f26d714c46b86c4188b3fd37c79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.348ex; height:2.343ex;" alt="{\displaystyle A\mathbf {x} \in \mathbb {R} ^{m}}" loading="lazy">. Conversely, any linear map between <a href="Finite-dimensional" class="mw-redirect" title="Finite-dimensional">finite-dimensional</a> vector spaces can be represented in this manner; see the <a href="#Matrices">§ Matrices</a>, below.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f:V\to W}</annotation>

</semantics>

</math><img src="./abdeb127675edefd619f13be240e53fea69292a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\textstyle f:V\to W}" loading="lazy"> is an <a href="Isometry" title="Isometry">isometry</a> between real <a href="Normed_space" class="mw-redirect" title="Normed space">normed spaces</a> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(0)=0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f(0)=0}</annotation>

</semantics>

</math><img src="./27ccedb9b9dff623a16340a136e140eea39f32b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.511ex; height:2.843ex;" alt="{\textstyle f(0)=0}" loading="lazy"> then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is a linear map. This result is not necessarily true for complex normed space.<a href="#cite_note-FOOTNOTEWilansky201321–26-8">[8]</a></li>

<li><a href="Derivative" title="Derivative">Differentiation</a> defines a linear map from the space of all differentiable functions to the space of all functions. It also defines a <a href="Linear_operator" class="mw-redirect" title="Linear operator">linear operator</a> on the space of all <a href="Smooth_function" class="mw-redirect" title="Smooth function">smooth functions</a> (a linear operator is a <a href="Linear_endomorphism" class="mw-redirect" title="Linear endomorphism">linear endomorphism</a>, that is, a linear map with the same <a href="Domain_of_a_function" title="Domain of a function">domain</a> and <a href="Codomain" title="Codomain">codomain</a>). Indeed, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}\left(af(x)+bg(x)\right)=a{\frac {df(x)}{dx}}+b{\frac {dg(x)}{dx}}.}">

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

</mrow>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}\left(af(x)+bg(x)\right)=a{\frac {df(x)}{dx}}+b{\frac {dg(x)}{dx}}.}</annotation>

</semantics>

</math></li>

<li>A definite <a href="Integral" title="Integral">integral</a> over some <a href="Interval_(mathematics)" title="Interval (mathematics)">interval</a> I is a linear map from the space of all real-valued integrable functions on I to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">. Indeed, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{u}^{v}\left(af(x)+bg(x)\right)dx=a\int _{u}^{v}f(x)dx+b\int _{u}^{v}g(x)dx.}">

</mrow>

</mrow>

</msubsup>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \int _{u}^{v}\left(af(x)+bg(x)\right)dx=a\int _{u}^{v}f(x)dx+b\int _{u}^{v}g(x)dx.}</annotation>

</semantics>

</math></li>

<li>An indefinite <a href="Integral" title="Integral">integral</a> (or <a href="Antiderivative" title="Antiderivative">antiderivative</a>) with a fixed integration starting point defines a linear map from the space of all real-valued integrable functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy"> to the space of all real-valued, differentiable functions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">. Without a fixed starting point, the antiderivative maps to the <a href="Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">quotient space</a> of the differentiable functions by the linear space of constant functions.</li>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> are finite-dimensional vector spaces over a field F, of respective dimensions m and n, then the function that maps linear maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f:V\to W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f:V\to W}</annotation>

</semantics>

</math><img src="./abdeb127675edefd619f13be240e53fea69292a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\textstyle f:V\to W}" loading="lazy"> to n × m matrices in the way described in <a href="#Matrices">§ Matrices</a> (below) is a linear map, and even a <a href="Linear_isomorphism" class="mw-redirect" title="Linear isomorphism">linear isomorphism</a>.</li>

<li>The <a href="Expected_value" title="Expected value">expected value</a> of a <a href="Random_variable#Definition" title="Random variable">random variable</a> (which is in fact a function, and as such an element of a vector space) is linear, as for random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> we have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[X+Y]=E[X]+E[Y]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E[X+Y]=E[X]+E[Y]}</annotation>

</semantics>

</math><img src="./ea2251900ec2b03db1d6f870336155a2a09ff7f1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.494ex; height:2.843ex;" alt="{\displaystyle E[X+Y]=E[X]+E[Y]}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E[aX]=aE[X]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E[aX]=aE[X]}</annotation>

</semantics>

</math><img src="./3e34453ad80cdf674f4d15fab3e8096be81af79a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.657ex; height:2.843ex;" alt="{\displaystyle E[aX]=aE[X]}" loading="lazy">, but the <a href="Variance" title="Variance">variance</a> of a random variable is not linear.</li></ul>

<div class="gallerytext">The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}</annotation>

</semantics>

</math><img src="./c06d7fc6902e01c04ce84cf57e41c7bd122f1556.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.294ex; height:3.009ex;" alt="{\textstyle f:\mathbb {R} ^{2}\to \mathbb {R} ^{2}}" loading="lazy"> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x,y)=(2x,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f(x,y)=(2x,y)}</annotation>

</semantics>

</math><img src="./dba4d012acc208d82dc8599aa34fa1823f4e6b33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.196ex; height:2.843ex;" alt="{\textstyle f(x,y)=(2x,y)}" loading="lazy"> is a linear map. This function scales the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle x}</annotation>

</semantics>

</math><img src="./d951e0f3b54b6a3d73bb9a0a005749046cbce781.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\textstyle x}" loading="lazy"> component of a vector by the factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle 2}</annotation>

</semantics>

</math><img src="./78ed0cd8140e5a15b6fcce83602df58458e0f3b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\textstyle 2}" loading="lazy">.</div>

</li>

<div class="gallerytext">The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x,y)=(2x,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f(x,y)=(2x,y)}</annotation>

</semantics>

</math><img src="./dba4d012acc208d82dc8599aa34fa1823f4e6b33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.196ex; height:2.843ex;" alt="{\textstyle f(x,y)=(2x,y)}" loading="lazy"> is additive: It does not matter whether vectors are first added and then mapped or whether they are mapped and finally added: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(\mathbf {a} +\mathbf {b} )=f(\mathbf {a} )+f(\mathbf {b} )}">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f(\mathbf {a} +\mathbf {b} )=f(\mathbf {a} )+f(\mathbf {b} )}</annotation>

</semantics>

</math><img src="./063ca24d48656cff63b970d6a536e55bb44c9fbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.612ex; height:2.843ex;" alt="{\textstyle f(\mathbf {a} +\mathbf {b} )=f(\mathbf {a} )+f(\mathbf {b} )}" loading="lazy"></div>

</li>

<div class="gallerytext">The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(x,y)=(2x,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f(x,y)=(2x,y)}</annotation>

</semantics>

</math><img src="./dba4d012acc208d82dc8599aa34fa1823f4e6b33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.196ex; height:2.843ex;" alt="{\textstyle f(x,y)=(2x,y)}" loading="lazy"> is homogeneous: It does not matter whether a vector is first scaled and then mapped or first mapped and then scaled: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f(\lambda \mathbf {a} )=\lambda f(\mathbf {a} )}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f(\lambda \mathbf {a} )=\lambda f(\mathbf {a} )}</annotation>

</semantics>

</math><img src="./47fc569c87a7898cddff2b904953e7b3044ecb2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.584ex; height:2.843ex;" alt="{\textstyle f(\lambda \mathbf {a} )=\lambda f(\mathbf {a} )}" loading="lazy"></div>

</li>

</ul>

<div class="mw-heading mw-heading3"><h3 id="Linear_extensions">Linear extensions</h3></div>

Often, a linear map is constructed by defining it on a subset of a vector space and then extending by linearity to the <a href="Linear_span" title="Linear span">linear span</a> of the domain.

Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> are vector spaces and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:S\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:S\to Y}</annotation>

</semantics>

</math><img src="./d3934b167cb8c46ba651326059aed50ddfd4d37a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.102ex; height:2.509ex;" alt="{\displaystyle f:S\to Y}" loading="lazy"> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> defined on some subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S\subseteq X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S\subseteq X.}</annotation>

</semantics>

</math><img src="./c586f8a77721c22346403a73aa1233c9cebd4f4d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.225ex; height:2.343ex;" alt="{\displaystyle S\subseteq X.}" loading="lazy">

Then a linear extension of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X,}</annotation>

</semantics>

</math><img src="./09ba32eeb405f7f5f2bac1eb12987c47d2fd42df.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.627ex; height:2.509ex;" alt="{\displaystyle X,}" loading="lazy"> if it exists, is a linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:X\to Y}</annotation>

</semantics>

</math><img src="./b81de3fd0af85cab5904d269f0e3f02088cea565.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.045ex; height:2.176ex;" alt="{\displaystyle F:X\to Y}" loading="lazy"> defined on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> that <a href="Extension_of_a_function" class="mw-redirect" title="Extension of a function">extends</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"><a href="#cite_note-9">[note 1]</a> (meaning that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)=f(s)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(s)=f(s)}</annotation>

</semantics>

</math><img src="./142eeec95286e4f13223ba5bc4cda26c5c5043ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.917ex; height:2.843ex;" alt="{\displaystyle F(s)=f(s)}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s\in S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s\in S}</annotation>

</semantics>

</math><img src="./acce52dffd84d073a24f4606a175da60148fd0c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.43ex; height:2.176ex;" alt="{\displaystyle s\in S}" loading="lazy">) and takes its values from the codomain of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f.}</annotation>

</semantics>

</math><img src="./ecb3ed2e17fa8f336dcc0fd4b3eddbfb02a50ef3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.925ex; height:2.509ex;" alt="{\displaystyle f.}" loading="lazy"><a href="#cite_note-FOOTNOTEKubrusly200157-10">[9]</a>

When the subset <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is a vector subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> then a (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">-valued) linear extension of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> to all of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is guaranteed to exist if (and only if) <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:S\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:S\to Y}</annotation>

</semantics>

</math><img src="./d3934b167cb8c46ba651326059aed50ddfd4d37a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.102ex; height:2.509ex;" alt="{\displaystyle f:S\to Y}" loading="lazy"> is a linear map.<a href="#cite_note-FOOTNOTEKubrusly200157-10">[9]</a> In particular, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> has a linear extension to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {span} S,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {span} S,}</annotation>

</semantics>

</math><img src="./5602d2b4577b2c054efb8209d905dbb0f2f0e23d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.197ex; height:2.509ex;" alt="{\displaystyle \operatorname {span} S,}" loading="lazy"> then it has a linear extension to all of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

The map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:S\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:S\to Y}</annotation>

</semantics>

</math><img src="./d3934b167cb8c46ba651326059aed50ddfd4d37a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.102ex; height:2.509ex;" alt="{\displaystyle f:S\to Y}" loading="lazy"> can be extended to a linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:\operatorname {span} S\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:\operatorname {span} S\to Y}</annotation>

</semantics>

</math><img src="./076b43c838f92c4b8039ca6d1ba462e7ab938d6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.615ex; height:2.509ex;" alt="{\displaystyle F:\operatorname {span} S\to Y}" loading="lazy"> if and only if whenever <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n>0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n>0}</annotation>

</semantics>

</math><img src="./27a6a5d982d54202a14f111cb8a49210501b2c96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.656ex; height:2.176ex;" alt="{\displaystyle n>0}" loading="lazy"> is an integer, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1},\ldots ,c_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{1},\ldots ,c_{n}}</annotation>

</semantics>

</math><img src="./22c1ec95175219801d58be5c63f155486225aa0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.465ex; height:2.009ex;" alt="{\displaystyle c_{1},\ldots ,c_{n}}" loading="lazy"> are scalars, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1},\ldots ,s_{n}\in S}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{1},\ldots ,s_{n}\in S}</annotation>

</semantics>

</math><img src="./d3fbf6ca885ac8fb5fff9508e6579f5ce23b31f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.972ex; height:2.509ex;" alt="{\displaystyle s_{1},\ldots ,s_{n}\in S}" loading="lazy"> are vectors such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=c_{1}s_{1}+\cdots +c_{n}s_{n},}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0=c_{1}s_{1}+\cdots +c_{n}s_{n},}</annotation>

</semantics>

</math><img src="./d9552f3743d5ad5f54da62c238736491c9fc8a74.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.052ex; height:2.509ex;" alt="{\displaystyle 0=c_{1}s_{1}+\cdots +c_{n}s_{n},}" loading="lazy"> then necessarily <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0=c_{1}f\left(s_{1}\right)+\cdots +c_{n}f\left(s_{n}\right).}">

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0=c_{1}f\left(s_{1}\right)+\cdots +c_{n}f\left(s_{n}\right).}</annotation>

</semantics>

</math><img src="./79734c7cb1c2f59665ce1224f9d2b5cc637c5ee5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.389ex; height:2.843ex;" alt="{\displaystyle 0=c_{1}f\left(s_{1}\right)+\cdots +c_{n}f\left(s_{n}\right).}" loading="lazy"><a href="#cite_note-FOOTNOTESchechter1996277–280-11">[10]</a>

If a linear extension of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:S\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:S\to Y}</annotation>

</semantics>

</math><img src="./d3934b167cb8c46ba651326059aed50ddfd4d37a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.102ex; height:2.509ex;" alt="{\displaystyle f:S\to Y}" loading="lazy"> exists then the linear extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:\operatorname {span} S\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:\operatorname {span} S\to Y}</annotation>

</semantics>

</math><img src="./076b43c838f92c4b8039ca6d1ba462e7ab938d6c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.615ex; height:2.509ex;" alt="{\displaystyle F:\operatorname {span} S\to Y}" loading="lazy"> is unique and

<mrow>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F\left(c_{1}s_{1}+\cdots c_{n}s_{n}\right)=c_{1}f\left(s_{1}\right)+\cdots +c_{n}f\left(s_{n}\right)}</annotation>

</semantics>

</math>

holds for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n,c_{1},\ldots ,c_{n},}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n,c_{1},\ldots ,c_{n},}</annotation>

</semantics>

</math><img src="./b7371e02ce9315d443fa2f7c51233076e6cf578e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.54ex; height:2.009ex;" alt="{\displaystyle n,c_{1},\ldots ,c_{n},}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{1},\ldots ,s_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s_{1},\ldots ,s_{n}}</annotation>

</semantics>

</math><img src="./5d60ff9326f87b9e18078832cee6aff971320175.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.632ex; height:2.009ex;" alt="{\displaystyle s_{1},\ldots ,s_{n}}" loading="lazy"> as above.<a href="#cite_note-FOOTNOTESchechter1996277–280-11">[10]</a>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> is linearly independent then every function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:S\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:S\to Y}</annotation>

</semantics>

</math><img src="./d3934b167cb8c46ba651326059aed50ddfd4d37a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.102ex; height:2.509ex;" alt="{\displaystyle f:S\to Y}" loading="lazy"> into any vector space has a linear extension to a (linear) map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \;\operatorname {span} S\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \;\operatorname {span} S\to Y}</annotation>

</semantics>

</math><img src="./ddfaa796e617b6e1b02cda7db5d3d2258a826c12.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.583ex; height:2.509ex;" alt="{\displaystyle \;\operatorname {span} S\to Y}" loading="lazy"> (the converse is also true).

For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X=\mathbb {R} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X=\mathbb {R} ^{2}}</annotation>

</semantics>

</math><img src="./e7927b9ff2cf803f94f24dfc058375f09ab7c370.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.811ex; height:2.676ex;" alt="{\displaystyle X=\mathbb {R} ^{2}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y=\mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y=\mathbb {R} }</annotation>

</semantics>

</math><img src="./2f56cb0c6e949f59fe9cf8f7fd4f01273319c4d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.55ex; height:2.176ex;" alt="{\displaystyle Y=\mathbb {R} }" loading="lazy"> then the assignment <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,0)\to -1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (1,0)\to -1}</annotation>

</semantics>

</math><img src="./21b751cb2bb5cef5ed1960648511e481542ddcfe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.753ex; height:2.843ex;" alt="{\displaystyle (1,0)\to -1}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,1)\to 2}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,1)\to 2}</annotation>

</semantics>

</math><img src="./db42d44628e5ebdbcd501b29cc805636cb6ae788.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.945ex; height:2.843ex;" alt="{\displaystyle (0,1)\to 2}" loading="lazy"> can be linearly extended from the linearly independent set of vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S:=\{(1,0),(0,1)\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S:=\{(1,0),(0,1)\}}</annotation>

</semantics>

</math><img src="./10046c83a9e556bb018ed380376a6ead62369ddf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.94ex; height:2.843ex;" alt="{\displaystyle S:=\{(1,0),(0,1)\}}" loading="lazy"> to a linear map on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {span} \{(1,0),(0,1)\}=\mathbb {R} ^{2}.}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {span} \{(1,0),(0,1)\}=\mathbb {R} ^{2}.}</annotation>

</semantics>

</math><img src="./6ecd75f93150744f6d25b2adb7dc0324e935a570.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.837ex; height:3.176ex;" alt="{\displaystyle \operatorname {span} \{(1,0),(0,1)\}=\mathbb {R} ^{2}.}" loading="lazy"> The unique linear extension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F:\mathbb {R} ^{2}\to \mathbb {R} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F:\mathbb {R} ^{2}\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./df3ae696ccec908679e0b0028a7ff40c3b778762.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.702ex; height:2.676ex;" alt="{\displaystyle F:\mathbb {R} ^{2}\to \mathbb {R} }" loading="lazy"> is the map that sends <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x,y)=x(1,0)+y(0,1)\in \mathbb {R} ^{2}}">

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (x,y)=x(1,0)+y(0,1)\in \mathbb {R} ^{2}}</annotation>

</semantics>

</math><img src="./3c5c79670eb1fa3a54a9ff6c3c3fe4a6d3e8edfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.662ex; height:3.176ex;" alt="{\displaystyle (x,y)=x(1,0)+y(0,1)\in \mathbb {R} ^{2}}" loading="lazy"> to

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(x,y)=x(-1)+y(2)=-x+2y.}</annotation>

</semantics>

</math>

Every (scalar-valued) <a href="Linear_functional" class="mw-redirect" title="Linear functional">linear functional</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> defined on a <a href="Vector_subspace" class="mw-redirect" title="Vector subspace">vector subspace</a> of a real or complex vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> has a linear extension to all of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X.}</annotation>

</semantics>

</math><img src="./5ba76c5a460c4a0bb1639a193bc1830f0a773e03.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.627ex; height:2.176ex;" alt="{\displaystyle X.}" loading="lazy">

Indeed, the <a href="Hahn%E2%80%93Banach_theorem" title="Hahn–Banach theorem">Hahn–Banach dominated extension theorem</a> even guarantees that when this linear functional <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is dominated by some given <a href="Seminorm" title="Seminorm">seminorm</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p:X\to \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p:X\to \mathbb {R} }</annotation>

</semantics>

</math><img src="./c8540db038c97641c58fdf2dbe344cc730d8b441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:10.468ex; height:2.509ex;" alt="{\displaystyle p:X\to \mathbb {R} }" loading="lazy"> (meaning that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |f(m)|\leq p(m)}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle |f(m)|\leq p(m)}</annotation>

</semantics>

</math><img src="./da70edf0a546825d259e88d00d9e1a0eeeb2cb56.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.54ex; height:2.843ex;" alt="{\displaystyle |f(m)|\leq p(m)}" loading="lazy"> holds for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" loading="lazy"> in the domain of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">) then there exists a linear extension to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> that is also dominated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p.}</annotation>

</semantics>

</math><img src="./88532f4eab1d4cef71ef96c0f8c98cac36fd9257.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.906ex; height:2.009ex;" alt="{\displaystyle p.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Matrices">Matrices</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Transformation_matrix" title="Transformation matrix">Transformation matrix</a></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> are <a href="Finite-dimensional" class="mw-redirect" title="Finite-dimensional">finite-dimensional</a> vector spaces and a <a href="Basis_of_a_vector_space" class="mw-redirect" title="Basis of a vector space">basis</a> is defined for each vector space, then every linear map from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy"> can be represented by a <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a>.<a href="#cite_note-12">[11]</a> This is useful because it allows concrete calculations. Matrices yield examples of linear maps: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> is a real <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation>

</semantics>

</math><img src="./12b23d207d23dd430b93320539abbb0bde84870d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}" loading="lazy"> matrix, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {x} )=A\mathbf {x} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\mathbf {x} )=A\mathbf {x} }</annotation>

</semantics>

</math><img src="./e8856fd6eb169daec22b0df95f5985d426e0b3c1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.751ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {x} )=A\mathbf {x} }" loading="lazy"> describes a linear map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}}">

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}}</annotation>

</semantics>

</math><img src="./cccdefd5f0e00fc2fa5fde2d8cbb039cc408a035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.864ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}\to \mathbb {R} ^{m}}" loading="lazy"> (see <a href="Euclidean_space" title="Euclidean space">Euclidean space</a>).

Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}\}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}\}}</annotation>

</semantics>

</math><img src="./333c9f5babb7e55da9f0a22cebaeca3bbc388e68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.598ex; height:2.843ex;" alt="{\displaystyle \{\mathbf {v} _{1},\ldots ,\mathbf {v} _{n}\}}" loading="lazy"> be a basis for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. Then every vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} \in V}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} \in V}</annotation>

</semantics>

</math><img src="./aeb70f294c05e671203c3b21e7f5fb3d0b9a0ced.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.039ex; height:2.176ex;" alt="{\displaystyle \mathbf {v} \in V}" loading="lazy"> is uniquely determined by the coefficients <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{1},\ldots ,c_{n}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{1},\ldots ,c_{n}}</annotation>

</semantics>

</math><img src="./22c1ec95175219801d58be5c63f155486225aa0d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.465ex; height:2.009ex;" alt="{\displaystyle c_{1},\ldots ,c_{n}}" loading="lazy"> in the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation>

</semantics>

</math><img src="./786849c765da7a84dbc3cce43e96aad58a5868dc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }" loading="lazy">:

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {v} =c_{1}\mathbf {v} _{1}+\cdots +c_{n}\mathbf {v} _{n}.}</annotation>

</semantics>

</math>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f:V\to W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f:V\to W}</annotation>

</semantics>

</math><img src="./abdeb127675edefd619f13be240e53fea69292a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\textstyle f:V\to W}" loading="lazy"> is a linear map,

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\mathbf {v} )=f(c_{1}\mathbf {v} _{1}+\cdots +c_{n}\mathbf {v} _{n})=c_{1}f(\mathbf {v} _{1})+\cdots +c_{n}f\left(\mathbf {v} _{n}\right),}</annotation>

</semantics>

</math>

which implies that the function f is entirely determined by the vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {v} _{1}),\ldots ,f(\mathbf {v} _{n})}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\mathbf {v} _{1}),\ldots ,f(\mathbf {v} _{n})}</annotation>

</semantics>

</math><img src="./19bf7f7c36e2b5347bec30ed830e8a3bf30a8130.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.449ex; height:2.843ex;" alt="{\displaystyle f(\mathbf {v} _{1}),\ldots ,f(\mathbf {v} _{n})}" loading="lazy">. Now let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\mathbf {w} _{1},\ldots ,\mathbf {w} _{m}\}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\mathbf {w} _{1},\ldots ,\mathbf {w} _{m}\}}</annotation>

</semantics>

</math><img src="./d540f5ca6b88d7f50b2a679f4f2b500c18d6fd99.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.095ex; height:2.843ex;" alt="{\displaystyle \{\mathbf {w} _{1},\ldots ,\mathbf {w} _{m}\}}" loading="lazy"> be a basis for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W}</annotation>

</semantics>

</math><img src="./54a9c4c547f4d6111f81946cad242b18298d70b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\displaystyle W}" loading="lazy">. Then we can represent each vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {v} _{j})}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\mathbf {v} _{j})}</annotation>

</semantics>

</math><img src="./b5f737cb0ad111b9aed28389f616680c76b2fe91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.409ex; height:3.009ex;" alt="{\displaystyle f(\mathbf {v} _{j})}" loading="lazy"> as

<mrow>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f\left(\mathbf {v} _{j}\right)=a_{1j}\mathbf {w} _{1}+\cdots +a_{mj}\mathbf {w} _{m}.}</annotation>

</semantics>

</math>

Thus, the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is entirely determined by the values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{ij}}</annotation>

</semantics>

</math><img src="./ebea6cd2813c330c798921a2894b358f7b643917.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.707ex; height:2.343ex;" alt="{\displaystyle a_{ij}}" loading="lazy">. If we put these values into an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m\times n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m\times n}</annotation>

</semantics>

</math><img src="./12b23d207d23dd430b93320539abbb0bde84870d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.276ex; height:1.676ex;" alt="{\displaystyle m\times n}" loading="lazy"> matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, then we can conveniently use it to compute the vector output of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> for any vector in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">. To get <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, every column <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is a vector

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}a_{1j}\\\vdots \\a_{mj}\end{pmatrix}}}</annotation>

</semantics>

</math>

corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {v} _{j})}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\mathbf {v} _{j})}</annotation>

</semantics>

</math><img src="./b5f737cb0ad111b9aed28389f616680c76b2fe91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.409ex; height:3.009ex;" alt="{\displaystyle f(\mathbf {v} _{j})}" loading="lazy"> as defined above. To define it more clearly, for some column <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy"> that corresponds to the mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {v} _{j})}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\mathbf {v} _{j})}</annotation>

</semantics>

</math><img src="./b5f737cb0ad111b9aed28389f616680c76b2fe91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.409ex; height:3.009ex;" alt="{\displaystyle f(\mathbf {v} _{j})}" loading="lazy">,

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {M} ={\begin{pmatrix}\ \cdots &a_{1j}&\cdots \ \\&\vdots &\\&a_{mj}&\end{pmatrix}}}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is the matrix of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy">. In other words, every column <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j=1,\ldots ,n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j=1,\ldots ,n}</annotation>

</semantics>

</math><img src="./c39c89dce6309ea91bb90112a6babe359d6ff7d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:11.819ex; height:2.509ex;" alt="{\displaystyle j=1,\ldots ,n}" loading="lazy"> has a corresponding vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\mathbf {v} _{j})}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(\mathbf {v} _{j})}</annotation>

</semantics>

</math><img src="./b5f737cb0ad111b9aed28389f616680c76b2fe91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.409ex; height:3.009ex;" alt="{\displaystyle f(\mathbf {v} _{j})}" loading="lazy"> whose coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{1j},\cdots ,a_{mj}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a_{1j},\cdots ,a_{mj}}</annotation>

</semantics>

</math><img src="./a7b69c5df086776004f3bf0d2bc3c9032d5a6751.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.722ex; height:2.343ex;" alt="{\displaystyle a_{1j},\cdots ,a_{mj}}" loading="lazy"> are the elements of column <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle j}</annotation>

</semantics>

</math><img src="./2f461e54f5c093e92a55547b9764291390f0b5d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" loading="lazy">. A single linear map may be represented by many matrices. This is because the values of the elements of a matrix depend on the bases chosen.

The matrices of a linear transformation can be represented visually:

<ol><li>Matrix for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle T}</annotation>

</semantics>

</math><img src="./db299e88e5485f250f4ba15530469c8c6080a8cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\textstyle T}" loading="lazy"> relative to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle B}</annotation>

</semantics>

</math><img src="./de0b47ffc21636dc2df68f6c793177a268f10e9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\textstyle B}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle A}</annotation>

</semantics>

</math><img src="./a118c6ad00742b3f5dccd2f0e74b5e369df6fd31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\textstyle A}" loading="lazy"></li>

<li>Matrix for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle T}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle T}</annotation>

</semantics>

</math><img src="./db299e88e5485f250f4ba15530469c8c6080a8cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\textstyle T}" loading="lazy"> relative to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle B'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle B'}</annotation>

</semantics>

</math><img src="./e1f60ed44f0ecd873daf5c8db8a9d9696c7a95d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.343ex;" alt="{\textstyle B'}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle A'}</annotation>

</semantics>

</math><img src="./02fbfa8c1b341407bf3fada3408fb66c78be3149.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.428ex; height:2.343ex;" alt="{\textstyle A'}" loading="lazy"></li>

<li>Transition matrix from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle B'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle B'}</annotation>

</semantics>

</math><img src="./e1f60ed44f0ecd873daf5c8db8a9d9696c7a95d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.343ex;" alt="{\textstyle B'}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle B}</annotation>

</semantics>

</math><img src="./de0b47ffc21636dc2df68f6c793177a268f10e9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\textstyle B}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle P}</annotation>

</semantics>

</math><img src="./038590207af1024a629c1a08c855e9ac46bf5610.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\textstyle P}" loading="lazy"></li>

<li>Transition matrix from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle B}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle B}</annotation>

</semantics>

</math><img src="./de0b47ffc21636dc2df68f6c793177a268f10e9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\textstyle B}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle B'}">

<msup>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle B'}</annotation>

</semantics>

</math><img src="./e1f60ed44f0ecd873daf5c8db8a9d9696c7a95d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.449ex; height:2.343ex;" alt="{\textstyle B'}" loading="lazy">: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle P^{-1}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle P^{-1}}</annotation>

</semantics>

</math><img src="./882153e03e0be618ee6351e09bf173319c5244f3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.154ex; height:2.509ex;" alt="{\textstyle P^{-1}}" loading="lazy"></li></ol>

Such that starting in the bottom left corner <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left[\mathbf {v} \right]_{B'}}">

<msub>

<mrow>

</mrow>

</mrow>

<msup>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \left[\mathbf {v} \right]_{B'}}</annotation>

</semantics>

</math><img src="./38ce4e66d088a57d8adb2aadc461213818bc3220.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.716ex; height:3.009ex;" alt="{\textstyle \left[\mathbf {v} \right]_{B'}}" loading="lazy"> and looking for the bottom right corner <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left[T\left(\mathbf {v} \right)\right]_{B'}}">

<msub>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<msup>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \left[T\left(\mathbf {v} \right)\right]_{B'}}</annotation>

</semantics>

</math><img src="./954ce89644d8b424263ec4dac91becfb6672d931.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.548ex; height:3.009ex;" alt="{\textstyle \left[T\left(\mathbf {v} \right)\right]_{B'}}" loading="lazy">, one would left-multiply—that is, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A'\left[\mathbf {v} \right]_{B'}=\left[T\left(\mathbf {v} \right)\right]_{B'}}">

<msup>

</msup>

<msub>

<mrow>

</mrow>

</mrow>

<msup>

</msup>

</mrow>

</msub>

<msub>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<msup>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle A'\left[\mathbf {v} \right]_{B'}=\left[T\left(\mathbf {v} \right)\right]_{B'}}</annotation>

</semantics>

</math><img src="./d40bbccc105af68b575ef57b62b1e1943c478aaa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.791ex; height:3.009ex;" alt="{\textstyle A'\left[\mathbf {v} \right]_{B'}=\left[T\left(\mathbf {v} \right)\right]_{B'}}" loading="lazy">. The equivalent method would be the "longer" method going clockwise from the same point such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left[\mathbf {v} \right]_{B'}}">

<msub>

<mrow>

</mrow>

</mrow>

<msup>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \left[\mathbf {v} \right]_{B'}}</annotation>

</semantics>

</math><img src="./38ce4e66d088a57d8adb2aadc461213818bc3220.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.716ex; height:3.009ex;" alt="{\textstyle \left[\mathbf {v} \right]_{B'}}" loading="lazy"> is left-multiplied with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle P^{-1}AP}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle P^{-1}AP}</annotation>

</semantics>

</math><img src="./5f599ab4304a233fa3389f59583109079effd806.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.643ex; height:2.509ex;" alt="{\textstyle P^{-1}AP}" loading="lazy">, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle P^{-1}AP\left[\mathbf {v} \right]_{B'}=\left[T\left(\mathbf {v} \right)\right]_{B'}}">

<msup>

</mrow>

</msup>

<msub>

<mrow>

</mrow>

</mrow>

<msup>

</msup>

</mrow>

</msub>

<msub>

<mrow>

<mrow>

<mrow>

</mrow>

</mrow>

</mrow>

<msup>

</msup>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle P^{-1}AP\left[\mathbf {v} \right]_{B'}=\left[T\left(\mathbf {v} \right)\right]_{B'}}</annotation>

</semantics>

</math><img src="./5c713b4830cf922213fe7907677ae35be5fffcfc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.006ex; height:3.176ex;" alt="{\textstyle P^{-1}AP\left[\mathbf {v} \right]_{B'}=\left[T\left(\mathbf {v} \right)\right]_{B'}}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Examples_in_two_dimensions">Examples in two dimensions</h3></div>

In two-<a href="Dimension" title="Dimension">dimensional</a> space R2 linear maps are described by 2 × 2 <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>. These are some examples:

<ul><li><a href="Rotation_(mathematics)" title="Rotation (mathematics)">rotation</a>

<ul><li>by 90 degrees counterclockwise: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}">

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{pmatrix}0&-1\\1&0\end{pmatrix}}}</annotation>

</semantics>

</math></li>

<li>by an angle θ counterclockwise: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{pmatrix}}}">

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{pmatrix}}}</annotation>

</semantics>

</math></li></ul></li>

<li><a href="Reflection_(mathematics)" title="Reflection (mathematics)">reflection</a>

<ul><li>through the x axis: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}">

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}</annotation>

</semantics>

</math></li>

<li>through the y axis: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{pmatrix}-1&0\\0&1\end{pmatrix}}}">

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{pmatrix}-1&0\\0&1\end{pmatrix}}}</annotation>

</semantics>

</math></li>

<li>through a line making an angle θ with the origin: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{pmatrix}\cos 2\theta &\sin 2\theta \\\sin 2\theta &-\cos 2\theta \end{pmatrix}}}">

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{pmatrix}\cos 2\theta &\sin 2\theta \\\sin 2\theta &-\cos 2\theta \end{pmatrix}}}</annotation>

</semantics>

</math></li></ul></li>

<li><a href="Scaling_(geometry)" title="Scaling (geometry)">scaling</a> by 2 in all directions: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{pmatrix}2&0\\0&2\end{pmatrix}}=2\mathbf {I} }">

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{pmatrix}2&0\\0&2\end{pmatrix}}=2\mathbf {I} }</annotation>

</semantics>

</math></li>

<li><a href="Shear_mapping" title="Shear mapping">horizontal shear mapping</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{pmatrix}1&m\\0&1\end{pmatrix}}}">

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{pmatrix}1&m\\0&1\end{pmatrix}}}</annotation>

</semantics>

</math></li>

<li>skew of the y axis by an angle θ: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{pmatrix}1&-\sin \theta \\0&\cos \theta \end{pmatrix}}}">

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{pmatrix}1&-\sin \theta \\0&\cos \theta \end{pmatrix}}}</annotation>

</semantics>

</math></li>

<li><a href="Squeeze_mapping" title="Squeeze mapping">squeeze mapping</a>: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{pmatrix}k&0\\0&{\frac {1}{k}}\end{pmatrix}}}">

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{pmatrix}k&0\\0&{\frac {1}{k}}\end{pmatrix}}}</annotation>

</semantics>

</math></li>

<li><a href="Projection_(linear_algebra)" title="Projection (linear algebra)">projection</a> onto the y axis: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {A} ={\begin{pmatrix}0&0\\0&1\end{pmatrix}}.}">

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathbf {A} ={\begin{pmatrix}0&0\\0&1\end{pmatrix}}.}</annotation>

</semantics>

</math></li></ul>

If a linear map is only composed of rotation, reflection, and/or uniform scaling, then the linear map is a <a href="Conformal_linear_transformation" title="Conformal linear transformation">conformal linear transformation</a>.

<div class="mw-heading mw-heading2"><h2 id="Vector_space_of_linear_maps">Vector space of linear maps</h2></div>

The composition of linear maps is linear: if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:V\to W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:V\to W}</annotation>

</semantics>

</math><img src="./574dffa1c85efaef6b6ef553ebd8ad9cf7f87fd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\displaystyle f:V\to W}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle g:W\to Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle g:W\to Z}</annotation>

</semantics>

</math><img src="./aa6d336769cb8e2e7f01807361a750577a9de618.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.783ex; height:2.509ex;" alt="{\textstyle g:W\to Z}" loading="lazy"> are linear, then so is their <a href="Relation_composition" class="mw-redirect" title="Relation composition">composition</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle g\circ f:V\to Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle g\circ f:V\to Z}</annotation>

</semantics>

</math><img src="./9c271c698ec5bf92d613adc6c108191b7255d3ee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.608ex; height:2.509ex;" alt="{\textstyle g\circ f:V\to Z}" loading="lazy">. It follows from this that the <a href="Class_(set_theory)" title="Class (set theory)">class</a> of all vector spaces over a given field K, together with K-linear maps as <a href="Morphism" title="Morphism">morphisms</a>, forms a <a href="Category_(mathematics)" title="Category (mathematics)">category</a>.

The <a href="Inverse_function" title="Inverse function">inverse</a> of a linear map, when defined, is again a linear map.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f_{1}:V\to W}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f_{1}:V\to W}</annotation>

</semantics>

</math><img src="./5580af80f2164ff883a70b212bf71f90a98687b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.967ex; height:2.509ex;" alt="{\textstyle f_{1}:V\to W}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f_{2}:V\to W}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f_{2}:V\to W}</annotation>

</semantics>

</math><img src="./d2aa07ed1e99afb386d5800d822e5868e6ead396.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.967ex; height:2.509ex;" alt="{\textstyle f_{2}:V\to W}" loading="lazy"> are linear, then so is their <a href="Pointwise" title="Pointwise">pointwise</a> sum <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}+f_{2}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f_{1}+f_{2}}</annotation>

</semantics>

</math><img src="./7875d43d94bfdb30c7a2a5d72fea4a6960bc8787.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.227ex; height:2.509ex;" alt="{\displaystyle f_{1}+f_{2}}" loading="lazy">, which is defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f_{1}+f_{2})(\mathbf {x} )=f_{1}(\mathbf {x} )+f_{2}(\mathbf {x} )}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (f_{1}+f_{2})(\mathbf {x} )=f_{1}(\mathbf {x} )+f_{2}(\mathbf {x} )}</annotation>

</semantics>

</math><img src="./145744cb4dbc4550b4d8efe2c0367cab34aa6487.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.023ex; height:2.843ex;" alt="{\displaystyle (f_{1}+f_{2})(\mathbf {x} )=f_{1}(\mathbf {x} )+f_{2}(\mathbf {x} )}" loading="lazy">.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f:V\to W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f:V\to W}</annotation>

</semantics>

</math><img src="./abdeb127675edefd619f13be240e53fea69292a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\textstyle f:V\to W}" loading="lazy"> is linear and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \alpha }</annotation>

</semantics>

</math><img src="./0d86dbd6183264b2f8569da1751380b173c7b185.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\textstyle \alpha }" loading="lazy"> is an element of the ground field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle K}</annotation>

</semantics>

</math><img src="./985dcc2532a2d4d91b9a9610139216c63cf832d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\textstyle K}" loading="lazy">, then the map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \alpha f}</annotation>

</semantics>

</math><img src="./92df229298e2efdd1b742042aa01ce30ca4a3c76.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.766ex; height:2.509ex;" alt="{\textstyle \alpha f}" loading="lazy">, defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (\alpha f)(\mathbf {x} )=\alpha (f(\mathbf {x} ))}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle (\alpha f)(\mathbf {x} )=\alpha (f(\mathbf {x} ))}</annotation>

</semantics>

</math><img src="./ac29e0d3be09de95b773567b3ebf4b93214b300e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.69ex; height:2.843ex;" alt="{\textstyle (\alpha f)(\mathbf {x} )=\alpha (f(\mathbf {x} ))}" loading="lazy">, is also linear.

Thus the set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\mathcal {L}}(V,W)}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle {\mathcal {L}}(V,W)}</annotation>

</semantics>

</math><img src="./1013af4653ccaa67039dcff4a9e2e0af7a780534.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.669ex; height:2.843ex;" alt="{\textstyle {\mathcal {L}}(V,W)}" loading="lazy"> of linear maps from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle V}</annotation>

</semantics>

</math><img src="./d67b50b7ba03a56fea637093cf80e12807852d19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\textstyle V}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle W}</annotation>

</semantics>

</math><img src="./e95737ee2530885a10b104e9cd5331077e1c88d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\textstyle W}" loading="lazy"> itself forms a vector space over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle K}</annotation>

</semantics>

</math><img src="./985dcc2532a2d4d91b9a9610139216c63cf832d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\textstyle K}" loading="lazy">,<a href="#cite_note-13">[12]</a> sometimes denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {Hom} (V,W)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {Hom} (V,W)}</annotation>

</semantics>

</math><img src="./6c02a7e07ec8b60afb76d96cdf915f88fc253d5c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.907ex; height:2.843ex;" alt="{\textstyle \operatorname {Hom} (V,W)}" loading="lazy">.<a href="#cite_note-14">[13]</a> Furthermore, in the case that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V=W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle V=W}</annotation>

</semantics>

</math><img src="./6e9fcec532c5ab11d7dd4f3a7ac83d951179c547.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.321ex; height:2.176ex;" alt="{\textstyle V=W}" loading="lazy">, this vector space, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {End} (V)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {End} (V)}</annotation>

</semantics>

</math><img src="./ca806cf0f1bbcf831f80a9e76bb303d9ab60eea2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.764ex; height:2.843ex;" alt="{\textstyle \operatorname {End} (V)}" loading="lazy">, is an <a href="Associative_algebra" title="Associative algebra">associative algebra</a> under <a href="Composition_of_maps" class="mw-redirect" title="Composition of maps">composition of maps</a>, since the composition of two linear maps is again a linear map, and the composition of maps is always associative. This case is discussed in more detail below.

Given again the finite-dimensional case, if bases have been chosen, then the composition of linear maps corresponds to the <a href="Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>, the addition of linear maps corresponds to the <a href="Matrix_addition" title="Matrix addition">matrix addition</a>, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.

<div class="mw-heading mw-heading3"><h3 id="Endomorphisms_and_automorphisms">Endomorphisms and automorphisms</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Endomorphism" title="Endomorphism">Endomorphism</a> and <a href="Automorphism" title="Automorphism">Automorphism</a></div>

A linear transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f:V\to V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f:V\to V}</annotation>

</semantics>

</math><img src="./d4f05c06fb794079c3b28a1aa91c3ff7e4090d7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.404ex; height:2.509ex;" alt="{\textstyle f:V\to V}" loading="lazy"> is an <a href="Endomorphism" title="Endomorphism">endomorphism</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle V}</annotation>

</semantics>

</math><img src="./d67b50b7ba03a56fea637093cf80e12807852d19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\textstyle V}" loading="lazy">; the set of all such endomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {End} (V)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {End} (V)}</annotation>

</semantics>

</math><img src="./ca806cf0f1bbcf831f80a9e76bb303d9ab60eea2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.764ex; height:2.843ex;" alt="{\textstyle \operatorname {End} (V)}" loading="lazy"> together with addition, composition and scalar multiplication as defined above forms an <a href="Associative_algebra" title="Associative algebra">associative algebra</a> with identity element over the field <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle K}</annotation>

</semantics>

</math><img src="./985dcc2532a2d4d91b9a9610139216c63cf832d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\textstyle K}" loading="lazy"> (and in particular a <a href="Ring_(algebra)" class="mw-redirect" title="Ring (algebra)">ring</a>). The multiplicative identity element of this algebra is the <a href="Identity_function" title="Identity function">identity map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {id} :V\to V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {id} :V\to V}</annotation>

</semantics>

</math><img src="./574028fa7c4de459a79fddd72a281420f140ad92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.065ex; height:2.176ex;" alt="{\textstyle \operatorname {id} :V\to V}" loading="lazy">.

An endomorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle V}</annotation>

</semantics>

</math><img src="./d67b50b7ba03a56fea637093cf80e12807852d19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\textstyle V}" loading="lazy"> that is also an <a href="Isomorphism" title="Isomorphism">isomorphism</a> is called an <a href="Automorphism" title="Automorphism">automorphism</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle V}</annotation>

</semantics>

</math><img src="./d67b50b7ba03a56fea637093cf80e12807852d19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\textstyle V}" loading="lazy">. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle V}</annotation>

</semantics>

</math><img src="./d67b50b7ba03a56fea637093cf80e12807852d19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\textstyle V}" loading="lazy"> forms a <a href="Group_(math)" class="mw-redirect" title="Group (math)">group</a>, the <a href="Automorphism_group" title="Automorphism group">automorphism group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle V}</annotation>

</semantics>

</math><img src="./d67b50b7ba03a56fea637093cf80e12807852d19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\textstyle V}" loading="lazy"> which is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {Aut} (V)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {Aut} (V)}</annotation>

</semantics>

</math><img src="./2d2d57ebb96657a5dfd20060f637baf09bf42b1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.537ex; height:2.843ex;" alt="{\textstyle \operatorname {Aut} (V)}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {GL} (V)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {GL} (V)}</annotation>

</semantics>

</math><img src="./58f4d6d756bd1541eefd853c705db15ae2c22769.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.874ex; height:2.843ex;" alt="{\textstyle \operatorname {GL} (V)}" loading="lazy">. Since the automorphisms are precisely those <a href="Endomorphisms" class="mw-redirect" title="Endomorphisms">endomorphisms</a> which possess inverses under composition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {Aut} (V)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {Aut} (V)}</annotation>

</semantics>

</math><img src="./2d2d57ebb96657a5dfd20060f637baf09bf42b1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.537ex; height:2.843ex;" alt="{\textstyle \operatorname {Aut} (V)}" loading="lazy"> is the group of <a href="Unit_(ring_theory)" title="Unit (ring theory)">units</a> in the ring <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {End} (V)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {End} (V)}</annotation>

</semantics>

</math><img src="./ca806cf0f1bbcf831f80a9e76bb303d9ab60eea2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.764ex; height:2.843ex;" alt="{\textstyle \operatorname {End} (V)}" loading="lazy">.

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle V}</annotation>

</semantics>

</math><img src="./d67b50b7ba03a56fea637093cf80e12807852d19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\textstyle V}" loading="lazy"> has finite dimension <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle n}</annotation>

</semantics>

</math><img src="./cc6e1f880981346a604257ebcacdef24c0aca2d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\textstyle n}" loading="lazy">, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {End} (V)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {End} (V)}</annotation>

</semantics>

</math><img src="./ca806cf0f1bbcf831f80a9e76bb303d9ab60eea2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.764ex; height:2.843ex;" alt="{\textstyle \operatorname {End} (V)}" loading="lazy"> is <a href="Isomorphism" title="Isomorphism">isomorphic</a> to the <a href="Associative_algebra" title="Associative algebra">associative algebra</a> of all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n\times n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle n\times n}</annotation>

</semantics>

</math><img src="./b6fbcac74eb679edde27a00a9f2a17a51f9ea354.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\textstyle n\times n}" loading="lazy"> matrices with entries in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle K}</annotation>

</semantics>

</math><img src="./985dcc2532a2d4d91b9a9610139216c63cf832d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\textstyle K}" loading="lazy">. The automorphism group of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle V}</annotation>

</semantics>

</math><img src="./d67b50b7ba03a56fea637093cf80e12807852d19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\textstyle V}" loading="lazy"> is <a href="Group_isomorphism" title="Group isomorphism">isomorphic</a> to the <a href="General_linear_group" title="General linear group">general linear group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {GL} (n,K)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {GL} (n,K)}</annotation>

</semantics>

</math><img src="./eacb8fb84b2808e33a6db91b4214648f3b218718.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.581ex; height:2.843ex;" alt="{\textstyle \operatorname {GL} (n,K)}" loading="lazy"> of all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle n\times n}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle n\times n}</annotation>

</semantics>

</math><img src="./b6fbcac74eb679edde27a00a9f2a17a51f9ea354.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\textstyle n\times n}" loading="lazy"> invertible matrices with entries in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle K}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle K}</annotation>

</semantics>

</math><img src="./985dcc2532a2d4d91b9a9610139216c63cf832d0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\textstyle K}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Kernel,_image_and_the_rank–nullity_theorem">Kernel, image and the rank–nullity theorem</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Kernel_(linear_algebra)" title="Kernel (linear algebra)">Kernel (linear algebra)</a>, <a href="Image_(mathematics)" title="Image (mathematics)">Image (mathematics)</a>, and <a href="Rank_of_a_matrix" class="mw-redirect" title="Rank of a matrix">Rank of a matrix</a></div>

If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f:V\to W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f:V\to W}</annotation>

</semantics>

</math><img src="./abdeb127675edefd619f13be240e53fea69292a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\textstyle f:V\to W}" loading="lazy"> is linear, we define the <a href="Kernel_(linear_operator)" class="mw-redirect" title="Kernel (linear operator)">kernel</a> and the <a href="Image_(mathematics)" title="Image (mathematics)">image</a> or <a href="Range_of_a_function" title="Range of a function">range</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f}</annotation>

</semantics>

</math><img src="./e1b77076edca76caf3331d0551d1645b8f678283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}" loading="lazy"> by

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mrow>

</mrow>

</mrow>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\ker(f)&=\{\,\mathbf {x} \in V:f(\mathbf {x} )=\mathbf {0} \,\}\\\operatorname {im} (f)&=\{\,\mathbf {w} \in W:\mathbf {w} =f(\mathbf {x} ),\mathbf {x} \in V\,\}\end{aligned}}}</annotation>

</semantics>

</math>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \ker(f)}</annotation>

</semantics>

</math><img src="./e3779641d69c82b84463c7e076186474cb26cdcf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.259ex; height:2.843ex;" alt="{\textstyle \ker(f)}" loading="lazy"> is a <a href="Linear_subspace" title="Linear subspace">subspace</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle V}</annotation>

</semantics>

</math><img src="./d67b50b7ba03a56fea637093cf80e12807852d19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\textstyle V}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {im} (f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {im} (f)}</annotation>

</semantics>

</math><img src="./52fceef68e297314302e5a700b8cdefaba490f19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.671ex; height:2.843ex;" alt="{\textstyle \operatorname {im} (f)}" loading="lazy"> is a subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle W}</annotation>

</semantics>

</math><img src="./e95737ee2530885a10b104e9cd5331077e1c88d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\textstyle W}" loading="lazy">. The following <a href="Dimension" title="Dimension">dimension</a> formula is known as the <a href="Rank%E2%80%93nullity_theorem" title="Rank–nullity theorem">rank–nullity theorem</a>:<a href="#cite_note-15">[14]</a>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \dim(\ker(f))+\dim(\operatorname {im} (f))=\dim(V).}</annotation>

</semantics>

</math>

The number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \dim(\operatorname {im} (f))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \dim(\operatorname {im} (f))}</annotation>

</semantics>

</math><img src="./70d54bd0869ba98395f292cecfa7c29e6235b84f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.355ex; height:2.843ex;" alt="{\textstyle \dim(\operatorname {im} (f))}" loading="lazy"> is also called the <a href="Rank_of_a_matrix" class="mw-redirect" title="Rank of a matrix">rank</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f}</annotation>

</semantics>

</math><img src="./e1b77076edca76caf3331d0551d1645b8f678283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}" loading="lazy"> and written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {rank} (f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {rank} (f)}</annotation>

</semantics>

</math><img src="./7622bf7e3a0c52c402b1fb1dab2e23004442fa64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.682ex; height:2.843ex;" alt="{\textstyle \operatorname {rank} (f)}" loading="lazy">, or sometimes, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \rho (f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \rho (f)}</annotation>

</semantics>

</math><img src="./fb5a24bfb61ff0956fdc208fbb10965dbf85918e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.29ex; height:2.843ex;" alt="{\textstyle \rho (f)}" loading="lazy">;<a href="#cite_note-:0-16">[15]</a><a href="#cite_note-:1-17">[16]</a> the number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \dim(\ker(f))}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \dim(\ker(f))}</annotation>

</semantics>

</math><img src="./f340be630d0250ae9e6e72c27b0897d6e13fd76c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.944ex; height:2.843ex;" alt="{\textstyle \dim(\ker(f))}" loading="lazy"> is called the <a href="Kernel_(matrix)" class="mw-redirect" title="Kernel (matrix)">nullity</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f}</annotation>

</semantics>

</math><img src="./e1b77076edca76caf3331d0551d1645b8f678283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}" loading="lazy"> and written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {null} (f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \operatorname {null} (f)}</annotation>

</semantics>

</math><img src="./8760687404a5df02862948431004cb1b4322e12b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.967ex; height:2.843ex;" alt="{\textstyle \operatorname {null} (f)}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \nu (f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \nu (f)}</annotation>

</semantics>

</math><img src="./53b60213c9c9ea3b78e94dae4c8b826f007ee6ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.32ex; height:2.843ex;" alt="{\textstyle \nu (f)}" loading="lazy">.<a href="#cite_note-:0-16">[15]</a><a href="#cite_note-:1-17">[16]</a> If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle V}</annotation>

</semantics>

</math><img src="./d67b50b7ba03a56fea637093cf80e12807852d19.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\textstyle V}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle W}</annotation>

</semantics>

</math><img src="./e95737ee2530885a10b104e9cd5331077e1c88d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.435ex; height:2.176ex;" alt="{\textstyle W}" loading="lazy"> are finite-dimensional, bases have been chosen and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f}</annotation>

</semantics>

</math><img src="./e1b77076edca76caf3331d0551d1645b8f678283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}" loading="lazy"> is represented by the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle A}</annotation>

</semantics>

</math><img src="./a118c6ad00742b3f5dccd2f0e74b5e369df6fd31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\textstyle A}" loading="lazy">, then the rank and nullity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f}</annotation>

</semantics>

</math><img src="./e1b77076edca76caf3331d0551d1645b8f678283.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\textstyle f}" loading="lazy"> are equal to the rank and nullity of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle A}</annotation>

</semantics>

</math><img src="./a118c6ad00742b3f5dccd2f0e74b5e369df6fd31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\textstyle A}" loading="lazy">, respectively.

<div class="mw-heading mw-heading2"><h2 id="Cokernel">Cokernel</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Cokernel" title="Cokernel">Cokernel</a></div>

A subtler invariant of a linear transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle f:V\to W}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle f:V\to W}</annotation>

</semantics>

</math><img src="./abdeb127675edefd619f13be240e53fea69292a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.052ex; height:2.509ex;" alt="{\textstyle f:V\to W}" loading="lazy"> is the <a href="Cokernel" title="Cokernel">cokernel</a>, which is defined as

<mi>coker</mi>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {coker} (f):=W/f(V)=W/\operatorname {im} (f).}</annotation>

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</math>

This is the dual notion to the kernel: just as the kernel is a subspace of the domain, the co-kernel is a <a href="Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">quotient space</a> of the target. Formally, one has the <a href="Exact_sequence" title="Exact sequence">exact sequence</a>

<mi>coker</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\to \ker(f)\to V\to W\to \operatorname {coker} (f)\to 0.}</annotation>

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</math>

These can be interpreted thus: given a linear equation f(v) = w to solve,

<ul><li>the kernel is the space of solutions to the homogeneous equation f(v) = 0, and its dimension is the number of <a href="Degrees_of_freedom" title="Degrees of freedom">degrees of freedom</a> in the space of solutions, if it is not empty;</li>

<li>the co-kernel is the space of <a href="https://en.wiktionary.org/wiki/constraint" class="extiw external" title="wikt:constraint">constraints</a> that the solutions must satisfy, and its dimension is the maximal number of independent constraints.</li></ul>

The dimension of the co-kernel and the dimension of the image (the rank) add up to the dimension of the target space. For finite dimensions, this means that the dimension of the quotient space W/f(V) is the dimension of the target space minus the dimension of the image.

As a simple example, consider the map f: R2 → R2, given by f(x, y) = (0, y). Then for an equation f(x, y) = (a, b) to have a solution, we must have a = 0 (one constraint), and in that case the solution space is (x, b) or equivalently stated, (0, b) + (x, 0), (one degree of freedom). The kernel may be expressed as the subspace (x, 0) < V: the value of x is the freedom in a solution – while the cokernel may be expressed via the map W → R, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (a,b)\mapsto (a)}">

</mstyle>

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<annotation encoding="application/x-tex">{\textstyle (a,b)\mapsto (a)}</annotation>

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</math><img src="./1f53be07dccb809fa9764339428f5796bbc27a89.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.724ex; height:2.843ex;" alt="{\textstyle (a,b)\mapsto (a)}" loading="lazy">: given a vector (a, b), the value of a is the obstruction to there being a solution.

An example illustrating the infinite-dimensional case is afforded by the map f: R∞ → R∞, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left\{a_{n}\right\}\mapsto \left\{b_{n}\right\}}">

<mrow>

<msub>

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</msub>

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<annotation encoding="application/x-tex">{\textstyle \left\{a_{n}\right\}\mapsto \left\{b_{n}\right\}}</annotation>

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</math><img src="./71a9ce4d48b1a2fa236c77c64071ba6095f8fab1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.928ex; height:2.843ex;" alt="{\textstyle \left\{a_{n}\right\}\mapsto \left\{b_{n}\right\}}" loading="lazy"> with b1 = 0 and bn + 1 = an for n > 0. Its image consists of all sequences with first element 0, and thus its cokernel consists of the classes of sequences with identical first element. Thus, whereas its kernel has dimension 0 (it maps only the zero sequence to the zero sequence), its co-kernel has dimension 1. Since the domain and the target space are the same, the rank and the dimension of the kernel add up to the same <a href="Cardinal_number#Cardinal_addition" title="Cardinal number">sum</a> as the rank and the dimension of the co-kernel (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \aleph _{0}+0=\aleph _{0}+1}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \aleph _{0}+0=\aleph _{0}+1}</annotation>

</semantics>

</math><img src="./75625b037575c0eb5dfceea93de905fe4f4d4e8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.053ex; height:2.509ex;" alt="{\textstyle \aleph _{0}+0=\aleph _{0}+1}" loading="lazy">), but in the infinite-dimensional case it cannot be inferred that the kernel and the co-kernel of an <a href="Endomorphism" title="Endomorphism">endomorphism</a> have the same dimension (0 ≠ 1). The reverse situation obtains for the map h: R∞ → R∞, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left\{a_{n}\right\}\mapsto \left\{c_{n}\right\}}">

<mrow>

<msub>

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</msub>

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<mrow>

<msub>

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<annotation encoding="application/x-tex">{\textstyle \left\{a_{n}\right\}\mapsto \left\{c_{n}\right\}}</annotation>

</semantics>

</math><img src="./1df94ee38b2273f0d4094821dd78b11426e44aea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.937ex; height:2.843ex;" alt="{\textstyle \left\{a_{n}\right\}\mapsto \left\{c_{n}\right\}}" loading="lazy"> with cn = an + 1. Its image is the entire target space, and hence its co-kernel has dimension 0, but since it maps all sequences in which only the first element is non-zero to the zero sequence, its kernel has dimension 1.

<div class="mw-heading mw-heading3"><h3 id="Index">Index</h3></div>

For a linear operator with finite-dimensional kernel and co-kernel, one may define index as:

<mi>coker</mi>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {ind} (f):=\dim(\ker(f))-\dim(\operatorname {coker} (f)),}</annotation>

</semantics>

</math>

namely the degrees of freedom minus the number of constraints.

For a transformation between finite-dimensional vector spaces, this is just the difference dim(V) − dim(W), by rank–nullity. This gives an indication of how many solutions or how many constraints one has: if mapping from a larger space to a smaller one, the map may be onto, and thus will have degrees of freedom even without constraints. Conversely, if mapping from a smaller space to a larger one, the map cannot be onto, and thus one will have constraints even without degrees of freedom.

The index of an operator is precisely the <a href="Euler_characteristic" title="Euler characteristic">Euler characteristic</a> of the 2-term complex 0 → V → W → 0. In <a href="Operator_theory" title="Operator theory">operator theory</a>, the index of <a href="Fredholm_operator" title="Fredholm operator">Fredholm operators</a> is an object of study, with a major result being the <a href="Atiyah%E2%80%93Singer_index_theorem" title="Atiyah–Singer index theorem">Atiyah–Singer index theorem</a>.<a href="#cite_note-18">[17]</a>

<div class="mw-heading mw-heading2"><h2 id="Algebraic_classifications_of_linear_transformations">Algebraic classifications of linear transformations</h2></div>

No classification of linear maps could be exhaustive. The following incomplete list enumerates some important classifications that do not require any additional structure on the vector space.

Let V and W denote vector spaces over a field F and let T: V → W be a linear map.

<div class="mw-heading mw-heading3"><h3 id="Monomorphism">Monomorphism</h3></div>

T is said to be <a href="Injective" class="mw-redirect" title="Injective">injective</a> or a <a href="Monomorphism" title="Monomorphism">monomorphism</a> if any of the following equivalent conditions are true:

<li>T is <a href="Monic_morphism" class="mw-redirect" title="Monic morphism">monic</a> or left-cancellable, which is to say, for any vector space U and any pair of linear maps R: U → V and S: U → V, the equation TR = TS implies R = S.</li>

<li>T is <a href="Inverse_(ring_theory)" class="mw-redirect" title="Inverse (ring theory)">left-invertible</a>, which is to say there exists a linear map S: W → V such that ST is the <a href="Identity_function" title="Identity function">identity map</a> on V.</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Epimorphism">Epimorphism</h3></div>

T is said to be <a href="Surjective" class="mw-redirect" title="Surjective">surjective</a> or an <a href="Epimorphism" title="Epimorphism">epimorphism</a> if any of the following equivalent conditions are true:

<li><a href="Cokernel" title="Cokernel">coker</a> T = {0W}</li>

<li>T is <a href="Epimorphism" title="Epimorphism">epic</a> or right-cancellable, which is to say, for any vector space U and any pair of linear maps R: W → U and S: W → U, the equation RT = ST implies R = S.</li>

<li>T is <a href="Inverse_(ring_theory)" class="mw-redirect" title="Inverse (ring theory)">right-invertible</a>, which is to say there exists a linear map S: W → V such that TS is the <a href="Identity_function" title="Identity function">identity map</a> on W.</li></ol>

<div class="mw-heading mw-heading3"><h3 id="Isomorphism">Isomorphism</h3></div>

T is said to be an <a href="Isomorphism" title="Isomorphism">isomorphism</a> if it is both left- and right-invertible. This is equivalent to T being both one-to-one and onto (a <a href="Bijection" title="Bijection">bijection</a> of sets) or also to T being both epic and monic, and so being a <a href="Bimorphism" class="mw-redirect" title="Bimorphism">bimorphism</a>.

If T: V → V is an endomorphism, then:

<ul><li>If, for some positive integer n, the n-th iterate of T, Tn, is identically zero, then T is said to be <a href="Nilpotent" title="Nilpotent">nilpotent</a>.</li>

<li>If T2 = T, then T is said to be <a href="Idempotent" class="mw-redirect" title="Idempotent">idempotent</a></li>

<li>If T = kI, where k is some scalar, then T is said to be a scaling transformation or scalar multiplication map; see <a href="Scalar_matrix" class="mw-redirect" title="Scalar matrix">scalar matrix</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Change_of_basis">Change of basis</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Basis_(linear_algebra)" title="Basis (linear algebra)">Basis (linear algebra)</a> and <a href="Change_of_basis" title="Change of basis">Change of basis</a></div>

Given a linear map which is an <a href="Endomorphism" title="Endomorphism">endomorphism</a> whose matrix is A, in the basis B of the space it transforms vector coordinates [u] as [v] = A[u]. As vectors change with the inverse of B (vectors coordinates are <a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">contravariant</a>) its inverse transformation is [v] = B[v'].

Substituting this in the first expression

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<annotation encoding="application/x-tex">{\displaystyle B\left[v'\right]=AB\left[u'\right]}</annotation>

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hence

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<annotation encoding="application/x-tex">{\displaystyle \left[v'\right]=B^{-1}AB\left[u'\right]=A'\left[u'\right].}</annotation>

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</math>

Therefore, the matrix in the new basis is A′ = B−1AB, being B the matrix of the given basis.

Therefore, linear maps are said to be 1-co- 1-contra-<a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">variant</a> objects, or type (1, 1) <a href="Tensor" title="Tensor">tensors</a>.

<div class="mw-heading mw-heading2"><h2 id="Continuity">Continuity</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Continuous_linear_operator" title="Continuous linear operator">Continuous linear operator</a> and <a href="Discontinuous_linear_map" title="Discontinuous linear map">Discontinuous linear map</a></div>

A linear transformation between <a href="Topological_vector_space" title="Topological vector space">topological vector spaces</a>, for example <a href="Normed_space" class="mw-redirect" title="Normed space">normed spaces</a>, may be <a href="Continuous_function_(topology)" class="mw-redirect" title="Continuous function (topology)">continuous</a>. If its domain and codomain are the same, it will then be a <a href="Continuous_linear_operator" title="Continuous linear operator">continuous linear operator</a>. A linear operator on a normed linear space is continuous if and only if it is <a href="Bounded_operator" title="Bounded operator">bounded</a>, for example, when the domain is finite-dimensional.<a href="#cite_note-19">[18]</a> An infinite-dimensional domain may have <a href="Discontinuous_linear_operator" class="mw-redirect" title="Discontinuous linear operator">discontinuous linear operators</a>.

An example of an unbounded, hence discontinuous, linear transformation is differentiation on the space of smooth functions equipped with the supremum norm (a function with small values can have a derivative with large values, while the derivative of 0 is 0). For a specific example, sin(nx)/n converges to 0, but its derivative cos(nx) does not, so differentiation is not continuous at 0 (and by a variation of this argument, it is not continuous anywhere).

<div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2></div>

A specific application of linear maps is for <a href="Geometric_transformation" title="Geometric transformation">geometric transformations</a>, such as those performed in <a href="Computer_graphics" title="Computer graphics">computer graphics</a>, where the translation, rotation and scaling of 2D or 3D objects is performed by the use of a <a href="Transformation_matrix" title="Transformation matrix">transformation matrix</a>. Linear mappings also are used as a mechanism for describing change: for example in calculus correspond to derivatives; or in relativity, used as a device to keep track of the local transformations of reference frames.

Another application of these transformations is in <a href="Compiler_optimizations" class="mw-redirect" title="Compiler optimizations">compiler optimizations</a> of nested-loop code, and in <a href="Parallelizing_compiler" class="mw-redirect" title="Parallelizing compiler">parallelizing compiler</a> techniques.

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<div class="side-box-text plainlist">Wikibooks has a book on the topic of: <a href="https://en.wikibooks.org/wiki/Linear_Algebra/Linear_Transformations" class="extiw external" title="wikibooks:Linear Algebra/Linear Transformations">Linear Algebra/Linear Transformations</a></div></div>

</div>

<ul><li><a href="Additive_map" title="Additive map">Additive map</a> – Z-module homomorphism</li>

<li><a href="Antilinear_map" title="Antilinear map">Antilinear map</a> – Conjugate homogeneous additive map</li>

<li><a href="Bent_function" title="Bent function">Bent function</a> – Special type of Boolean function</li>

<li><a href="Bounded_operator" title="Bounded operator">Bounded operator</a></li>

<li><a href="Cauchy's_functional_equation" title="Cauchy's functional equation">Cauchy's functional equation</a> – Functional equation</li>

<li><a href="Continuous_linear_operator" title="Continuous linear operator">Continuous linear operator</a> – Function between topological vector spaces</li>

<li><a href="Linear_functional" class="mw-redirect" title="Linear functional">Linear functional</a> – Linear map from a vector space to its field of scalarsPages displaying short descriptions of redirect targets</li>

<li><a href="Linear_isometry" class="mw-redirect" title="Linear isometry">Linear isometry</a> – Distance-preserving mathematical transformationPages displaying short descriptions of redirect targets</li>

<li><a href="Category_of_matrices" title="Category of matrices">Category of matrices</a></li>

<li><a href="Quasilinearization" title="Quasilinearization">Quasilinearization</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

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<li id="cite_note-1"><a href="#cite_ref-1">^</a> "Linear transformations of V into V are often called linear operators on V." <a href="#CITEREFRudin1976">Rudin 1976</a>, p. 207

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> Let V and W be two real vector spaces. A mapping a from V into W Is called a 'linear mapping' or 'linear transformation' or 'linear operator' [...] from V into W, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a(\mathbf {u} +\mathbf {v} )=a\mathbf {u} +a\mathbf {v} }">

</mrow>

</mrow>

</mrow>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle a(\mathbf {u} +\mathbf {v} )=a\mathbf {u} +a\mathbf {v} }</annotation>

</semantics>

</math><img src="./8228dbd9913a19b1642793d7e41ff0155c6a4329.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.071ex; height:2.843ex;" alt="{\textstyle a(\mathbf {u} +\mathbf {v} )=a\mathbf {u} +a\mathbf {v} }" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {u} ,\mathbf {v} \in V}">

</mrow>

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<annotation encoding="application/x-tex">{\textstyle \mathbf {u} ,\mathbf {v} \in V}</annotation>

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</math><img src="./f683aa60560de6fabc20a35c25e37d485cd6c1ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.558ex; height:2.509ex;" alt="{\textstyle \mathbf {u} ,\mathbf {v} \in V}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a(\lambda \mathbf {u} )=\lambda a\mathbf {u} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle a(\lambda \mathbf {u} )=\lambda a\mathbf {u} }</annotation>

</semantics>

</math><img src="./a50b87cd4c01f3c46d95d8e9511143a4f615eb92.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.048ex; height:2.843ex;" alt="{\textstyle a(\lambda \mathbf {u} )=\lambda a\mathbf {u} }" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} \in V}">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {u} \in V}</annotation>

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</math><img src="./f9760c4f376b6a157440323c062e96eb26f03d7b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.113ex; height:2.176ex;" alt="{\displaystyle \mathbf {u} \in V}" loading="lazy"> and all real λ. <a href="#CITEREFBronshteinSemendyayev2004">Bronshtein & Semendyayev 2004</a>, p. 316

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, p. 14 Here are some properties of linear mappings <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Lambda :X\to Y}">

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<annotation encoding="application/x-tex">{\textstyle \Lambda :X\to Y}</annotation>

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</math><img src="./c4a5ade851a427047e34eea3fe0b71f21f7140a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.918ex; height:2.176ex;" alt="{\textstyle \Lambda :X\to Y}" loading="lazy"> whose proofs are so easy that we omit them; it is assumed that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A\subset X}">

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<annotation encoding="application/x-tex">{\textstyle A\subset X}</annotation>

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</math><img src="./ceb5a0b3b24e085c222d4e58366a979f692aa97b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.822ex; height:2.176ex;" alt="{\textstyle A\subset X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle B\subset Y}">

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<annotation encoding="application/x-tex">{\textstyle B\subset Y}</annotation>

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</math><img src="./a4d99602a2a7e1f142f3d4909b019f8810d8713c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.636ex; height:2.176ex;" alt="{\textstyle B\subset Y}" loading="lazy">:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Lambda 0=0.}</annotation>

</semantics>

</math><img src="./f2e4eeb6ccfcb9ad2de35e3887d7b7040d8c6c13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.683ex; height:2.176ex;" alt="{\textstyle \Lambda 0=0.}" loading="lazy"></li><li>If A is a subspace (or a <a href="Convex_set" title="Convex set">convex set</a>, or a <a href="Balanced_set" title="Balanced set">balanced set</a>) the same is true of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Lambda (A)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Lambda (A)}</annotation>

</semantics>

</math><img src="./3d7d605d1df780e24a44a0f0ba6a4cccb7a197c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.165ex; height:2.843ex;" alt="{\textstyle \Lambda (A)}" loading="lazy"></li><li>If B is a subspace (or a convex set, or a balanced set) the same is true of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Lambda ^{-1}(B)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Lambda ^{-1}(B)}</annotation>

</semantics>

</math><img src="./a667bcca2e645e4b3f6713039da2884871d34c1c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.519ex; height:3.009ex;" alt="{\textstyle \Lambda ^{-1}(B)}" loading="lazy"></li><li>In particular, the set: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda ^{-1}(\{0\})=\{\mathbf {x} \in X:\Lambda \mathbf {x} =0\}={N}(\Lambda )}">

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Lambda ^{-1}(\{0\})=\{\mathbf {x} \in X:\Lambda \mathbf {x} =0\}={N}(\Lambda )}</annotation>

</semantics>

</math> is a subspace of X, called the null space of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Lambda }</annotation>

</semantics>

</math><img src="./229e74dffa35f9483f374a8e72dd682a2d3b7438.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\textstyle \Lambda }" loading="lazy">.</li></ol></div>

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, p. 14. Suppose now that X and Y are vector spaces over the same scalar field. A mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Lambda :X\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Lambda :X\to Y}</annotation>

</semantics>

</math><img src="./c4a5ade851a427047e34eea3fe0b71f21f7140a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.918ex; height:2.176ex;" alt="{\textstyle \Lambda :X\to Y}" loading="lazy"> is said to be linear if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Lambda (\alpha \mathbf {x} +\beta \mathbf {y} )=\alpha \Lambda \mathbf {x} +\beta \Lambda \mathbf {y} }">

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Lambda (\alpha \mathbf {x} +\beta \mathbf {y} )=\alpha \Lambda \mathbf {x} +\beta \Lambda \mathbf {y} }</annotation>

</semantics>

</math><img src="./1ea8550867761916ae758c347c301ac97a777ce6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.711ex; height:2.843ex;" alt="{\textstyle \Lambda (\alpha \mathbf {x} +\beta \mathbf {y} )=\alpha \Lambda \mathbf {x} +\beta \Lambda \mathbf {y} }" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {x} ,\mathbf {y} \in X}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {x} ,\mathbf {y} \in X}</annotation>

</semantics>

</math><img src="./9106cf7a7f7928e9c71d88169f0d47c634cd56ce.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.677ex; height:2.509ex;" alt="{\textstyle \mathbf {x} ,\mathbf {y} \in X}" loading="lazy"> and all scalars <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \alpha }</annotation>

</semantics>

</math><img src="./0d86dbd6183264b2f8569da1751380b173c7b185.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\textstyle \alpha }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \beta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \beta }</annotation>

</semantics>

</math><img src="./a77bfb138e56b5b44f6c8c4ce32a05449d1573d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\textstyle \beta }" loading="lazy">. Note that one often writes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Lambda \mathbf {x} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Lambda \mathbf {x} }</annotation>

</semantics>

</math><img src="./2af05ae06ea12ae10a59c52a7cb8a16d02fcac21.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.024ex; height:2.176ex;" alt="{\textstyle \Lambda \mathbf {x} }" loading="lazy">, rather than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Lambda (\mathbf {x} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Lambda (\mathbf {x} )}</annotation>

</semantics>

</math><img src="./34dc38adb693a76a3e27f6c2ca255c5db78aed9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.833ex; height:2.843ex;" alt="{\textstyle \Lambda (\mathbf {x} )}" loading="lazy">, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Lambda }</annotation>

</semantics>

</math><img src="./229e74dffa35f9483f374a8e72dd682a2d3b7438.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\textstyle \Lambda }" loading="lazy"> is linear.

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> <a href="#CITEREFRudin1976">Rudin 1976</a>, p. 206. A mapping A of a vector space X into a vector space Y is said to be a linear transformation if: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A\left(\mathbf {x} _{1}+\mathbf {x} _{2}\right)=A\mathbf {x} _{1}+A\mathbf {x} _{2},\ A(c\mathbf {x} )=cA\mathbf {x} }">

<mrow>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle A\left(\mathbf {x} _{1}+\mathbf {x} _{2}\right)=A\mathbf {x} _{1}+A\mathbf {x} _{2},\ A(c\mathbf {x} )=cA\mathbf {x} }</annotation>

</semantics>

</math><img src="./9fbfabeff5be27ea5477569588e9c2221c11eae0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.91ex; height:2.843ex;" alt="{\textstyle A\left(\mathbf {x} _{1}+\mathbf {x} _{2}\right)=A\mathbf {x} _{1}+A\mathbf {x} _{2},\ A(c\mathbf {x} )=cA\mathbf {x} }" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {x} ,\mathbf {x} _{1},\mathbf {x} _{2}\in X}">

</mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {x} ,\mathbf {x} _{1},\mathbf {x} _{2}\in X}</annotation>

</semantics>

</math><img src="./4193cb4f4e261460d39f02cb6fdca8e3b8d13a25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.23ex; height:2.509ex;" alt="{\textstyle \mathbf {x} ,\mathbf {x} _{1},\mathbf {x} _{2}\in X}" loading="lazy"> and all scalars c. Note that one often writes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A\mathbf {x} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle A\mathbf {x} }</annotation>

</semantics>

</math><img src="./91141e5884b9ccbef922f0ecc0dd598c8f05dd5e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.154ex; height:2.176ex;" alt="{\textstyle A\mathbf {x} }" loading="lazy"> instead of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A(\mathbf {x} )}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle A(\mathbf {x} )}</annotation>

</semantics>

</math><img src="./5af6bd8c47e7c932538dfc6cd5dbad849824ac1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.963ex; height:2.843ex;" alt="{\textstyle A(\mathbf {x} )}" loading="lazy"> if A is linear.

</li>

<li id="cite_note-6"><a href="#cite_ref-6">^</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, p. 14.

Linear mappings of X onto its scalar field are called linear functionals.

</li>

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://math.stackexchange.com/questions/62789/what-does-linear-mean-in-linear-algebra">"terminology - What does 'linear' mean in Linear Algebra?"</a>. Mathematics Stack Exchange. Retrieved 2021-02-17.</cite>

</li>

<li id="cite_note-FOOTNOTEWilansky201321–26-8"><a href="#cite_ref-FOOTNOTEWilansky201321–26_8-0">^</a> <a href="#CITEREFWilansky2013">Wilansky 2013</a>, pp. 21–26.

</li>

<li id="cite_note-FOOTNOTEKubrusly200157-10">^ <a href="#cite_ref-FOOTNOTEKubrusly200157_10-0">a</a> <a href="#cite_ref-FOOTNOTEKubrusly200157_10-1">b</a> <a href="#CITEREFKubrusly2001">Kubrusly 2001</a>, p. 57.

</li>

<li id="cite_note-FOOTNOTESchechter1996277–280-11">^ <a href="#cite_ref-FOOTNOTESchechter1996277–280_11-0">a</a> <a href="#cite_ref-FOOTNOTESchechter1996277–280_11-1">b</a> <a href="#CITEREFSchechter1996">Schechter 1996</a>, pp. 277–280.

</li>

<li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFRudin1976">Rudin 1976</a>, p. 210

Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left\{\mathbf {x} _{1},\ldots ,\mathbf {x} _{n}\right\}}">

<mrow>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \left\{\mathbf {x} _{1},\ldots ,\mathbf {x} _{n}\right\}}</annotation>

</semantics>

</math><img src="./4ac4d529f3388811c88fc38df6f2ce58a1b480c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.598ex; height:2.843ex;" alt="{\textstyle \left\{\mathbf {x} _{1},\ldots ,\mathbf {x} _{n}\right\}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \left\{\mathbf {y} _{1},\ldots ,\mathbf {y} _{m}\right\}}">

<mrow>

<mrow>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \left\{\mathbf {y} _{1},\ldots ,\mathbf {y} _{m}\right\}}</annotation>

</semantics>

</math><img src="./0282480646428ba56bd3aff1813d3e186293a08c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.054ex; height:2.843ex;" alt="{\textstyle \left\{\mathbf {y} _{1},\ldots ,\mathbf {y} _{m}\right\}}" loading="lazy"> are bases of vector spaces X and Y, respectively. Then every <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A\in L(X,Y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle A\in L(X,Y)}</annotation>

</semantics>

</math><img src="./e5193b0d2623e66fb05c8d678f698492c9bd07a8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.763ex; height:2.843ex;" alt="{\textstyle A\in L(X,Y)}" loading="lazy"> determines a set of numbers <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a_{i,j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle a_{i,j}}</annotation>

</semantics>

</math><img src="./e2d0b0ded3acabfd1fb9fbe9bf2499026a46fad3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.164ex; height:2.343ex;" alt="{\textstyle a_{i,j}}" loading="lazy"> such that

<msub>

</mrow>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A\mathbf {x} _{j}=\sum _{i=1}^{m}a_{i,j}\mathbf {y} _{i}\quad (1\leq j\leq n).}</annotation>

</semantics>

</math>

It is convenient to represent these numbers in a rectangular array of m rows and n columns, called an m by n matrix:

<mrow>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

</mtd>

</mtr>

<mtr>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

<mtd>

</mtd>

<mtd>

<msub>

</mrow>

</msub>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [A]={\begin{bmatrix}a_{1,1}&a_{1,2}&\ldots &a_{1,n}\\a_{2,1}&a_{2,2}&\ldots &a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\a_{m,1}&a_{m,2}&\ldots &a_{m,n}\end{bmatrix}}}</annotation>

</semantics>

</math>

Observe that the coordinates <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle a_{i,j}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle a_{i,j}}</annotation>

</semantics>

</math><img src="./e2d0b0ded3acabfd1fb9fbe9bf2499026a46fad3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.164ex; height:2.343ex;" alt="{\textstyle a_{i,j}}" loading="lazy"> of the vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A\mathbf {x} _{j}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle A\mathbf {x} _{j}}</annotation>

</semantics>

</math><img src="./9b522484c1f7e160d561489469a9c112e834fd34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.064ex; height:2.843ex;" alt="{\textstyle A\mathbf {x} _{j}}" loading="lazy"> (with respect to the basis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \{\mathbf {y} _{1},\ldots ,\mathbf {y} _{m}\}}">

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \{\mathbf {y} _{1},\ldots ,\mathbf {y} _{m}\}}</annotation>

</semantics>

</math><img src="./361925f84c9ecc0e9a9cef578b24678f4d78fe3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.054ex; height:2.843ex;" alt="{\textstyle \{\mathbf {y} _{1},\ldots ,\mathbf {y} _{m}\}}" loading="lazy">) appear in the jth column of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [A]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle [A]}</annotation>

</semantics>

</math><img src="./efc7c42d7e9f789e2b3384a4f718faddeb0f2119.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.037ex; height:2.843ex;" alt="{\textstyle [A]}" loading="lazy">. The vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle A\mathbf {x} _{j}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle A\mathbf {x} _{j}}</annotation>

</semantics>

</math><img src="./9b522484c1f7e160d561489469a9c112e834fd34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.064ex; height:2.843ex;" alt="{\textstyle A\mathbf {x} _{j}}" loading="lazy"> are therefore sometimes called the column vectors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [A]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle [A]}</annotation>

</semantics>

</math><img src="./efc7c42d7e9f789e2b3384a4f718faddeb0f2119.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.037ex; height:2.843ex;" alt="{\textstyle [A]}" loading="lazy">. With this terminology, the range of A is spanned by the column vectors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle [A]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle [A]}</annotation>

</semantics>

</math><img src="./efc7c42d7e9f789e2b3384a4f718faddeb0f2119.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.037ex; height:2.843ex;" alt="{\textstyle [A]}" loading="lazy">.

</li>

<li id="cite_note-13"><a href="#cite_ref-13">^</a> <a href="#CITEREFAxler2015">Axler (2015)</a> p. 52, § 3.3

</li>

<li id="cite_note-14"><a href="#cite_ref-14">^</a> <a href="#CITEREFTu2011">Tu (2011)</a>, p. 19, § 3.1

</li>

<li id="cite_note-15"><a href="#cite_ref-15">^</a> <a href="#CITEREFHornJohnson2013">Horn & Johnson 2013</a>, 0.2.3 Vector spaces associated with a matrix or linear transformation, p. 6

</li>

<li id="cite_note-:0-16">^ <a href="#cite_ref-:0_16-0">a</a> <a href="#cite_ref-:0_16-1">b</a> <a href="#CITEREFKatznelsonKatznelson2008">Katznelson & Katznelson (2008)</a> p. 52, § 2.5.1

</li>

<li id="cite_note-:1-17">^ <a href="#cite_ref-:1_17-0">a</a> <a href="#cite_ref-:1_17-1">b</a> <a href="#CITEREFHalmos1974">Halmos (1974)</a> p. 90, § 50

</li>

<li id="cite_note-18"><a href="#cite_ref-18">^</a> <cite id="CITEREFNistor2001" class="citation cs2">Nistor, Victor (2001) [1994], <a rel="nofollow" class="external text" href="https://www.encyclopediaofmath.org/index.php?title=Index_theory">"Index theory"</a>, <a href="Encyclopedia_of_Mathematics" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="European_Mathematical_Society" title="European Mathematical Society">EMS Press</a></cite>: "The main question in index theory is to provide index formulas for classes of Fredholm operators ... Index theory has become a subject on its own only after M. F. Atiyah and I. Singer published their index theorems"

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <a href="#CITEREFRudin1991">Rudin 1991</a>, p. 15

1.18 Theorem Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Lambda }</annotation>

</semantics>

</math><img src="./229e74dffa35f9483f374a8e72dd682a2d3b7438.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\textstyle \Lambda }" loading="lazy"> be a linear functional on a topological vector space X. Assume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Lambda \mathbf {x} \neq 0}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Lambda \mathbf {x} \neq 0}</annotation>

</semantics>

</math><img src="./1a976dd0a7808a6566f1897e6342e42c0a9b240a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.285ex; height:2.676ex;" alt="{\textstyle \Lambda \mathbf {x} \neq 0}" loading="lazy"> for some <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \mathbf {x} \in X}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \mathbf {x} \in X}</annotation>

</semantics>

</math><img src="./d258ec728ecceac7e214c9cac1a56cf683e15812.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.232ex; height:2.176ex;" alt="{\textstyle \mathbf {x} \in X}" loading="lazy">. Then each of the following four properties implies the other three:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Lambda }</annotation>

</semantics>

</math><img src="./229e74dffa35f9483f374a8e72dd682a2d3b7438.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\textstyle \Lambda }" loading="lazy"> is continuous</li><li>The null space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle N(\Lambda )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle N(\Lambda )}</annotation>

</semantics>

</math><img src="./c7c012b3b906681ca885fa0ce5dda9c33ca0cd0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.486ex; height:2.843ex;" alt="{\textstyle N(\Lambda )}" loading="lazy"> is closed.</li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle N(\Lambda )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle N(\Lambda )}</annotation>

</semantics>

</math><img src="./c7c012b3b906681ca885fa0ce5dda9c33ca0cd0b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.486ex; height:2.843ex;" alt="{\textstyle N(\Lambda )}" loading="lazy"> is not dense in X.</li><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \Lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle \Lambda }</annotation>

</semantics>

</math><img src="./229e74dffa35f9483f374a8e72dd682a2d3b7438.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\textstyle \Lambda }" loading="lazy"> is bounded in some neighbourhood V of 0.</li></ol></div>

</li>

</ol></div></div>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> is said to <a href="Extension_of_a_function" class="mw-redirect" title="Extension of a function">extend</a> another map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> if when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f}</annotation>

</semantics>

</math><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"> is defined at a point <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s,}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s,}</annotation>

</semantics>

</math><img src="./e5748cdb81bf00075de8e7e6828c343687513830.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.737ex; height:2.009ex;" alt="{\displaystyle s,}" loading="lazy"> then so is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F}</annotation>

</semantics>

</math><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F(s)=f(s).}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F(s)=f(s).}</annotation>

</semantics>

</math><img src="./ded63a97975cf243f85bb972cb17786524508e47.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.564ex; height:2.843ex;" alt="{\displaystyle F(s)=f(s).}" loading="lazy">

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="Bibliography">Bibliography</h2></div>

<ul><li><cite id="CITEREFAxler2015" class="citation book cs1"><a href="Sheldon_Axler" title="Sheldon Axler">Axler, Sheldon Jay</a> (2015). Linear Algebra Done Right (3rd ed.). <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-3-319-11079-0</bdi>.</cite></li>

<li><cite id="CITEREFBronshteinSemendyayev2004" class="citation book cs1">Bronshtein, I. N.; Semendyayev, K. A. (2004). <a href="Bronshtein_and_Semendyayev" title="Bronshtein and Semendyayev">Handbook of Mathematics</a> (4th ed.). New York: Springer-Verlag. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>3-540-43491-7</bdi>.</cite></li>

<li><cite id="CITEREFHalmos1974" class="citation book cs1"><a href="Paul_Halmos" title="Paul Halmos">Halmos, Paul Richard</a> (1974) [1958]. Finite-Dimensional Vector Spaces (2nd ed.). <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-90093-4</bdi>.</cite></li>

<li><cite id="CITEREFHornJohnson2013" class="citation book cs1">Horn, Roger A.; Johnson, Charles R. (2013). Matrix Analysis (Second ed.). <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-521-83940-2</bdi>.</cite></li>

<li><cite id="CITEREFKatznelsonKatznelson2008" class="citation book cs1"><a href="Yitzhak_Katznelson" title="Yitzhak Katznelson">Katznelson, Yitzhak</a>; Katznelson, Yonatan R. (2008). A (Terse) Introduction to Linear Algebra. <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-4419-9</bdi>.</cite></li>

<li><cite id="CITEREFKubrusly2001" class="citation book cs1">Kubrusly, Carlos (2001). Elements of operator theory. Boston: Birkhäuser. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4757-3328-0</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/754555941">754555941</a>.</cite></li>

<li><cite id="CITEREFLang1987" class="citation cs2"><a href="Serge_Lang" title="Serge Lang">Lang, Serge</a> (1987), Linear Algebra (Third ed.), New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-387-96412-6</bdi></cite></li>

<li><cite id="CITEREFRudin1973" class="citation book cs1"><a href="Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1973). <a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys00rudi">Functional Analysis</a>. International Series in Pure and Applied Mathematics. Vol. 25 (First ed.). New York, NY: <a href="McGraw-Hill_Science/Engineering/Math" class="mw-redirect" title="McGraw-Hill Science/Engineering/Math">McGraw-Hill Science/Engineering/Math</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780070542259</bdi>.</cite></li>

<li><cite id="CITEREFRudin1976" class="citation book cs1"><a href="Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1976). <a rel="nofollow" class="external text" href="https://archive.org/details/PrinciplesOfMathematicalAnalysis">Principles of Mathematical Analysis</a>. Walter Rudin Student Series in Advanced Mathematics (3rd ed.). New York: McGraw–Hill. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-054235-8</bdi>.</cite></li>

<li><cite id="CITEREFRudin1991" class="citation book cs1"><a href="Walter_Rudin" title="Walter Rudin">Rudin, Walter</a> (1991). <a rel="nofollow" class="external text" href="https://archive.org/details/functionalanalys00rudi">Functional Analysis</a>. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: <a href="McGraw-Hill_Science/Engineering/Math" class="mw-redirect" title="McGraw-Hill Science/Engineering/Math">McGraw-Hill Science/Engineering/Math</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-054236-5</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/21163277">21163277</a>.</cite></li>

<li><cite id="CITEREFSchaeferWolff1999" class="citation book cs1"><a href="Helmut_H._Schaefer" title="Helmut H. Schaefer">Schaefer, Helmut H.</a>; Wolff, Manfred P. (1999). Topological Vector Spaces. <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">GTM</a>. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4612-7155-0</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/840278135">840278135</a>.</cite></li>

<li><cite id="CITEREFSchechter1996" class="citation book cs1"><a href="Eric_Schechter" title="Eric Schechter">Schechter, Eric</a> (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-622760-4</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/175294365">175294365</a>.</cite></li>

<li><cite id="CITEREFSwartz1992" class="citation book cs1">Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8247-8643-4</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/24909067">24909067</a>.</cite></li>

<li><cite id="CITEREFTu2011" class="citation book cs1"><a href="Loring_W._Tu" title="Loring W. Tu">Tu, Loring W.</a> (2011). An Introduction to Manifolds (2nd ed.). <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-4419-9</bdi>.</cite></li>

<li><cite id="CITEREFWilansky2013" class="citation book cs1"><a href="Albert_Wilansky" title="Albert Wilansky">Wilansky, Albert</a> (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-486-49353-4</bdi>. <a href="OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/849801114">849801114</a>.</cite></li></ul>

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</style><div id="Linear_algebra379" style="font-size:114%;margin:0 4em"><a href="Linear_algebra" title="Linear algebra">Linear algebra</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li><a href="Outline_of_linear_algebra" title="Outline of linear algebra">Outline</a></li>

<li><a href="Glossary_of_linear_algebra" title="Glossary of linear algebra">Glossary</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Scalar_(mathematics)" title="Scalar (mathematics)">Scalar</a></li>

<li><a href="Euclidean_vector" title="Euclidean vector">Vector</a></li>

<li><a href="Vector_space" title="Vector space">Vector space</a></li>

<li><a href="Scalar_multiplication" title="Scalar multiplication">Scalar multiplication</a></li>

<li><a href="Vector_projection" title="Vector projection">Vector projection</a></li>

<li><a href="Linear_span" title="Linear span">Linear span</a></li>

<li><a href="Projection_(linear_algebra)" title="Projection (linear algebra)">Linear projection</a></li>

<li><a href="Linear_independence" title="Linear independence">Linear independence</a></li>

<li><a href="Linear_combination" title="Linear combination">Linear combination</a></li>

<li><a href="Multilinear_map" title="Multilinear map">Multilinear map</a></li>

<li><a href="Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li>

<li><a href="Change_of_basis" title="Change of basis">Change of basis</a></li>

<li><a href="Row_and_column_vectors" title="Row and column vectors">Row and column vectors</a></li>

<li><a href="Row_and_column_spaces" title="Row and column spaces">Row and column spaces</a></li>

<li><a href="Kernel_(linear_algebra)" title="Kernel (linear algebra)">Kernel</a></li>

<li><a href="Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">Eigenvalues and eigenvectors</a></li>

<li><a href="Transpose" title="Transpose">Transpose</a></li>

<li><a href="System_of_linear_equations" title="System of linear equations">Linear equations</a></li></ul>

</div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><a href="Euclidean_space" title="Euclidean space"></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Matrix_(mathematics)" title="Matrix (mathematics)">Matrices</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Block_matrix" title="Block matrix">Block</a></li>

<li><a href="Matrix_decomposition" title="Matrix decomposition">Decomposition</a></li>

<li><a href="Invertible_matrix" title="Invertible matrix">Invertible</a></li>

<li><a href="Minor_(linear_algebra)" title="Minor (linear algebra)">Minor</a></li>

<li><a href="Matrix_multiplication" title="Matrix multiplication">Multiplication</a></li>

<li><a href="Transformation_matrix" title="Transformation matrix">Transformation</a></li>

<li><a href="Cramer's_rule" title="Cramer's rule">Cramer's rule</a></li>

<li><a href="Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a></li>

<li><a href="Productive_matrix" title="Productive matrix">Productive matrix</a></li>

<li><a href="Gram_matrix" title="Gram matrix">Gram matrix</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Bilinear_map" title="Bilinear map">Bilinear</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Orthogonality" title="Orthogonality">Orthogonality</a></li>

<li><a href="Dot_product" title="Dot product">Dot product</a></li>

<li><a href="Hadamard_product_(matrices)" title="Hadamard product (matrices)">Hadamard product</a></li>

<li><a href="Inner_product_space" title="Inner product space">Inner product space</a></li>

<li><a href="Outer_product" title="Outer product">Outer product</a></li>

<li><a href="Kronecker_product" title="Kronecker product">Kronecker product</a></li>

<li><a href="Gram%E2%80%93Schmidt_process" title="Gram–Schmidt process">Gram–Schmidt process</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Determinant" title="Determinant">Determinant</a></li>

<li><a href="Cross_product" title="Cross product">Cross product</a></li>

<li><a href="Triple_product" title="Triple product">Triple product</a></li>

<li><a href="Seven-dimensional_cross_product" title="Seven-dimensional cross product">Seven-dimensional cross product</a></li>

<li><a href="Geometric_algebra" title="Geometric algebra">Geometric algebra</a></li>

<li><a href="Exterior_algebra" title="Exterior algebra">Exterior algebra</a></li>

<li><a href="Bivector" title="Bivector">Bivector</a></li>

<li><a href="Multivector" title="Multivector">Multivector</a></li>

<li><a href="Tensor" title="Tensor">Tensor</a></li>

<li><a href="Outermorphism" title="Outermorphism">Outermorphism</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Vector_space" title="Vector space">Vector space</a> constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Direct_sum_of_modules#Construction_for_two_vector_spaces" title="Direct sum of modules">Direct sum</a></li>

<li><a href="Function_space#In_linear_algebra" title="Function space">Function space</a></li>

<li><a href="Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">Quotient</a></li>

<li><a href="Linear_subspace" title="Linear subspace">Subspace</a></li>

<li><a href="Tensor_product" title="Tensor product">Tensor product</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Numerical_linear_algebra" title="Numerical linear algebra">Numerical</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Floating-point_arithmetic" title="Floating-point arithmetic">Floating-point</a></li>

<li><a href="Numerical_stability" title="Numerical stability">Numerical stability</a></li>

<li><a href="Basic_Linear_Algebra_Subprograms" title="Basic Linear Algebra Subprograms">Basic Linear Algebra Subprograms</a></li>

<li><a href="Sparse_matrix" title="Sparse matrix">Sparse matrix</a></li>

<li><a href="Comparison_of_linear_algebra_libraries" title="Comparison of linear algebra libraries">Comparison of linear algebra libraries</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div>

<ul><li> Category</li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Tensors176" style="padding:3px"><table class="nowraplinks hlist mw-collapsible mw-collapsed navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Tensors176" style="font-size:114%;margin:0 4em"><a href="Tensor" title="Tensor">Tensors</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="Glossary_of_tensor_theory" title="Glossary of tensor theory">Glossary of tensor theory</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Scope</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><a href="Mathematics" title="Mathematics">Mathematics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Coordinate_system" title="Coordinate system">Coordinate system</a></li>

<li><a href="Differential_geometry" title="Differential geometry">Differential geometry</a></li>

<li><a href="Dyadics" title="Dyadics">Dyadic algebra</a></li>

<li><a href="Euclidean_geometry" title="Euclidean geometry">Euclidean geometry</a></li>

<li><a href="Exterior_calculus" class="mw-redirect" title="Exterior calculus">Exterior calculus</a></li>

<li><a href="Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></li>

<li><a href="Tensor_algebra" title="Tensor algebra">Tensor algebra</a></li>

<li><a href="Tensor_calculus" class="mw-redirect" title="Tensor calculus">Tensor calculus</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;"><div class="hlist"><ul><li><a href="Physics" title="Physics">Physics</a></li><li><a href="Engineering" title="Engineering">Engineering</a></li></ul></div></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Computer_vision" title="Computer vision">Computer vision</a></li>

<li><a href="Continuum_mechanics" title="Continuum mechanics">Continuum mechanics</a></li>

<li><a href="Electromagnetism" title="Electromagnetism">Electromagnetism</a></li>

<li><a href="General_relativity" title="General relativity">General relativity</a></li>

<li><a href="Transport_phenomena" title="Transport phenomena">Transport phenomena</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notation</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Abstract_index_notation" title="Abstract index notation">Abstract index notation</a></li>

<li><a href="Einstein_notation" title="Einstein notation">Einstein notation</a></li>

<li><a href="Index_notation" title="Index notation">Index notation</a></li>

<li><a href="Multi-index_notation" title="Multi-index notation">Multi-index notation</a></li>

<li><a href="Penrose_graphical_notation" title="Penrose graphical notation">Penrose graphical notation</a></li>

<li><a href="Ricci_calculus" title="Ricci calculus">Ricci calculus</a></li>

<li><a href="Tetrad_(index_notation)" class="mw-redirect" title="Tetrad (index notation)">Tetrad (index notation)</a></li>

<li><a href="Van_der_Waerden_notation" title="Van der Waerden notation">Van der Waerden notation</a></li>

<li><a href="Voigt_notation" title="Voigt notation">Voigt notation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Tensor definitions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Tensor_(intrinsic_definition)" title="Tensor (intrinsic definition)">Tensor (intrinsic definition)</a></li>

<li><a href="Tensor_field" title="Tensor field">Tensor field</a></li>

<li><a href="Tensor_density" title="Tensor density">Tensor density</a></li>

<li><a href="Tensors_in_curvilinear_coordinates" title="Tensors in curvilinear coordinates">Tensors in curvilinear coordinates</a></li>

<li><a href="Mixed_tensor" title="Mixed tensor">Mixed tensor</a></li>

<li><a href="Antisymmetric_tensor" title="Antisymmetric tensor">Antisymmetric tensor</a></li>

<li><a href="Symmetric_tensor" title="Symmetric tensor">Symmetric tensor</a></li>

<li><a href="Tensor_operator" title="Tensor operator">Tensor operator</a></li>

<li><a href="Tensor_bundle" title="Tensor bundle">Tensor bundle</a></li>

<li><a href="Two-point_tensor" title="Two-point tensor">Two-point tensor</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Operation_(mathematics)" title="Operation (mathematics)">Operations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Covariant_derivative" title="Covariant derivative">Covariant derivative</a></li>

<li><a href="Exterior_covariant_derivative" title="Exterior covariant derivative">Exterior covariant derivative</a></li>

<li><a href="Exterior_derivative" title="Exterior derivative">Exterior derivative</a></li>

<li><a href="Exterior_product" class="mw-redirect" title="Exterior product">Exterior product</a></li>

<li><a href="Hodge_star_operator" title="Hodge star operator">Hodge star operator</a></li>

<li><a href="Lie_derivative" title="Lie derivative">Lie derivative</a></li>

<li><a href="Raising_and_lowering_indices" class="mw-redirect" title="Raising and lowering indices">Raising and lowering indices</a></li>

<li><a href="Symmetrization" title="Symmetrization">Symmetrization</a></li>

<li><a href="Tensor_contraction" title="Tensor contraction">Tensor contraction</a></li>

<li><a href="Tensor_product" title="Tensor product">Tensor product</a></li>

<li><a href="Transpose" title="Transpose">Transpose</a> (2nd-order tensors)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related abstractions</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Affine_connection" title="Affine connection">Affine connection</a></li>

<li><a href="Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li>

<li><a href="Cartan_formalism_(physics)" class="mw-redirect" title="Cartan formalism (physics)">Cartan formalism (physics)</a></li>

<li><a href="Connection_form" title="Connection form">Connection form</a></li>

<li><a href="Covariance_and_contravariance_of_vectors" title="Covariance and contravariance of vectors">Covariance and contravariance of vectors</a></li>

<li><a href="Differential_form" title="Differential form">Differential form</a></li>

<li><a href="Dimension" title="Dimension">Dimension</a></li>

<li><a href="Exterior_form" class="mw-redirect" title="Exterior form">Exterior form</a></li>

<li><a href="Fiber_bundle" title="Fiber bundle">Fiber bundle</a></li>

<li><a href="Geodesic" title="Geodesic">Geodesic</a></li>

<li><a href="Levi-Civita_connection" title="Levi-Civita connection">Levi-Civita connection</a></li>

<li><a href="Manifold" title="Manifold">Manifold</a></li>

<li><a href="Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a></li>

<li><a href="Multivector" title="Multivector">Multivector</a></li>

<li><a href="Pseudotensor" title="Pseudotensor">Pseudotensor</a></li>

<li><a href="Spinor" title="Spinor">Spinor</a></li>

<li><a href="Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">Vector</a></li>

<li><a href="Vector_space" title="Vector space">Vector space</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Notable tensors</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Mathematics</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Kronecker_delta" title="Kronecker delta">Kronecker delta</a></li>

<li><a href="Levi-Civita_symbol" title="Levi-Civita symbol">Levi-Civita symbol</a></li>

<li><a href="Metric_tensor" title="Metric tensor">Metric tensor</a></li>

<li><a href="Nonmetricity_tensor" title="Nonmetricity tensor">Nonmetricity tensor</a></li>

<li><a href="Ricci_curvature" title="Ricci curvature">Ricci curvature</a></li>

<li><a href="Riemann_curvature_tensor" title="Riemann curvature tensor">Riemann curvature tensor</a></li>

<li><a href="Torsion_tensor" title="Torsion tensor">Torsion tensor</a></li>

<li><a href="Weyl_tensor" title="Weyl tensor">Weyl tensor</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%;font-weight:normal;">Physics</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Moment_of_inertia#Inertia_tensor" title="Moment of inertia">Moment of inertia</a></li>

<li><a href="Angular_momentum#Angular_momentum_in_relativistic_mechanics" title="Angular momentum">Angular momentum tensor</a></li>

<li><a href="Spin_tensor" title="Spin tensor">Spin tensor</a></li>

<li><a href="Cauchy_stress_tensor" title="Cauchy stress tensor">Cauchy stress tensor</a></li>

<li><a href="Stress%E2%80%93energy_tensor" title="Stress–energy tensor">stress–energy tensor</a></li>

<li><a href="Einstein_tensor" title="Einstein tensor">Einstein tensor</a></li>

<li><a href="Electromagnetic_tensor" title="Electromagnetic tensor">EM tensor</a></li>

<li><a href="Gluon_field_strength_tensor" title="Gluon field strength tensor">Gluon field strength tensor</a></li>

<li><a href="Metric_tensor_(general_relativity)" title="Metric tensor (general relativity)">Metric tensor (GR)</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Mathematician" title="Mathematician">Mathematicians</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="%C3%89lie_Cartan" title="Élie Cartan">Élie Cartan</a></li>

<li><a href="Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a></li>

<li><a href="Elwin_Bruno_Christoffel" title="Elwin Bruno Christoffel">Elwin Bruno Christoffel</a></li>

<li><a href="Albert_Einstein" title="Albert Einstein">Albert Einstein</a></li>

<li><a href="Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a></li>

<li><a href="Carl_Friedrich_Gauss" title="Carl Friedrich Gauss">Carl Friedrich Gauss</a></li>

<li><a href="Hermann_Grassmann" title="Hermann Grassmann">Hermann Grassmann</a></li>

<li><a href="Tullio_Levi-Civita" title="Tullio Levi-Civita">Tullio Levi-Civita</a></li>

<li><a href="Gregorio_Ricci-Curbastro" title="Gregorio Ricci-Curbastro">Gregorio Ricci-Curbastro</a></li>

<li><a href="Bernhard_Riemann" title="Bernhard Riemann">Bernhard Riemann</a></li>

<li><a href="Jan_Arnoldus_Schouten" title="Jan Arnoldus Schouten">Jan Arnoldus Schouten</a></li>

<li><a href="Woldemar_Voigt" title="Woldemar Voigt">Woldemar Voigt</a></li>

<li><a href="Hermann_Weyl" title="Hermann Weyl">Hermann Weyl</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Function330" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Function330" style="font-size:114%;margin:0 4em"><a href="Function_(mathematics)" title="Function (mathematics)">Function</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="History_of_the_function_concept" title="History of the function concept">History</a></li>

<li><a href="List_of_mathematical_functions" title="List of mathematical functions">List of specific functions</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Types by domain and codomain</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Classes/properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Constant_function" title="Constant function">Constant</a></li>

<li><a href="Identity_function" title="Identity function">Identity</a></li>

<li><a href="Polynomial" title="Polynomial">Polynomial</a></li>

<li><a href="Rational_function" title="Rational function">Rational</a></li>

<li><a href="Algebraic_function" title="Algebraic function">Algebraic</a></li>

<li><a href="Analytic_function" title="Analytic function">Analytic</a></li>

<li><a href="Smooth_function" class="mw-redirect" title="Smooth function">Smooth</a></li>

<li><a href="Continuous_function" title="Continuous function">Continuous</a></li>

<li><a href="Measurable_function" title="Measurable function">Measurable</a></li>

<li><a href="Injective_function" title="Injective function">Injective</a></li>

<li><a href="Surjective_function" title="Surjective function">Surjective</a></li>

<li><a href="Bijection" title="Bijection">Bijective</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Restriction_(mathematics)" title="Restriction (mathematics)">Restriction</a></li>

<li><a href="Function_composition" title="Function composition">Composition</a></li>

<li><a href="Inverse_function" title="Inverse function">Inverse</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Generalizations</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Relation_(mathematics)" title="Relation (mathematics)">Relation</a> (<a href="Binary_relation" title="Binary relation">Binary relation</a>)</li>

<li><a href="Set-valued_function" title="Set-valued function">Set-valued</a></li>

<li><a href="Multivalued_function" title="Multivalued function">Multivalued</a></li>

<li><a href="Partial_function" title="Partial function">Partial</a></li>

<li><a href="Implicit_function" title="Implicit function">Implicit</a></li>

<li><a href="Function_space" title="Function space">Space</a></li>

<li><a href="Higher-order_function" title="Higher-order function">Higher-order</a></li>

<li><a href="Morphism" title="Morphism">Morphism</a></li>

<li><a href="Functor" title="Functor">Functor</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li> Category</li></ul>

</div></td></tr></tbody></table></div>

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</style></div><div role="navigation" class="navbox authority-control" aria-labelledby="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q207643#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1194" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Authority_control_databases_frameless&#124;text-top&#124;10px&#124;alt=Edit_this_at_Wikidata&#124;link=https&#58;//www.wikidata.org/wiki/Q207643#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1194" style="font-size:114%;margin:0 4em">Authority control databases </div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">National</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://d-nb.info/gnd/4167700-6">Germany</a></li><li><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85077178">United States</a></li><li><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11944511p">France</a></li><li><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11944511p">BnF data</a></li><li><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007529341405171">Israel</a></li></ul></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"><ul><li><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/b6f38c2e-0976-4a29-9e62-99177e29e90e">Yale LUX</a></li></ul></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

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