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<title>Linearity</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Linearity</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">"Linear" redirects here. For other uses, see <a href="Linear_(disambiguation)" class="mw-disambig" title="Linear (disambiguation)">Linear (disambiguation)</a>.</div>

<div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="Lineage_(disambiguation)" class="mw-redirect mw-disambig" title="Lineage (disambiguation)">Lineage</a>.</div>

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<p>In mathematics, the term <i><b>linear</b></i> is used in two distinct senses for two different properties:

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<ul><li>linearity of a <i><a href="Function_(mathematics)" title="Function (mathematics)">function</a></i> (or <i><a href="Mapping_(mathematics)" class="mw-redirect" title="Mapping (mathematics)">mapping</a></i>);</li>

<li>linearity of a <i><a href="Polynomial" title="Polynomial">polynomial</a></i>.</li></ul>

<p>An example of a linear function is the function defined by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=(ax,bx)}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)=(ax,bx)}</annotation>

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</math></span><img src="./ab83755bcaeef69cce2e5a8e0cc8d2bb110e78ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.246ex; height:2.843ex;" alt="{\displaystyle f(x)=(ax,bx)}" loading="lazy"></span> that maps the real line to a line in the <a href="Euclidean_plane" title="Euclidean plane">Euclidean plane</a> <b>R</b><sup>2</sup> that passes through the origin. An example of a linear polynomial in the variables <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X,}">

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

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</math></span><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"></span> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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

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</math></span><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"></span> and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

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

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</math></span><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy"></span> is <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle aX+bY+cZ+d.}">

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<annotation encoding="application/x-tex">{\displaystyle aX+bY+cZ+d.}</annotation>

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</math></span><img src="./ef862db1aff4bf344263610586bbd74d041bbb33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.052ex; height:2.343ex;" alt="{\displaystyle aX+bY+cZ+d.}" loading="lazy"></span>

</p><p>Linearity of a mapping is closely related to <i><a href="Proportionality_(mathematics)" title="Proportionality (mathematics)">proportionality</a></i>. Examples in <a href="Physics" title="Physics">physics</a> include the linear relationship of <a href="Voltage" title="Voltage">voltage</a> and <a href="Electric_current" title="Electric current">current</a> in an <a href="Electrical_conductor" title="Electrical conductor">electrical conductor</a> (<a href="Ohm's_law" title="Ohm's law">Ohm's law</a>), and the relationship of <a href="Mass" title="Mass">mass</a> and <a href="Weight" title="Weight">weight</a>. By contrast, more complicated relationships, such as between <a href="Velocity" title="Velocity">velocity</a> and <a href="Kinetic_energy" title="Kinetic energy">kinetic energy</a>, are <i><a href="Nonlinear_system" title="Nonlinear system">nonlinear</a></i>.

</p><p>Generalized for functions in more than one <a href="Dimension_(mathematics)" class="mw-redirect" title="Dimension (mathematics)">dimension</a>, linearity means the property of a function of being compatible with <a href="Addition" title="Addition">addition</a> and <a href="Scale_analysis_(mathematics)" title="Scale analysis (mathematics)">scaling</a>, also known as the <a href="Superposition_principle" title="Superposition principle">superposition principle</a>.

</p><p>Linearity of a polynomial means that its <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> is less than two. The use of the term for polynomials stems from the fact that the <a href="Graph_of_a_function" title="Graph of a function">graph</a> of a polynomial in one variable is a straight <a href="Line_(geometry)" title="Line (geometry)">line</a>. In the term "<a href="Linear_equation" title="Linear equation">linear equation</a>", the word refers to the linearity of the polynomials involved.

</p><p>Because a function such as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)=ax+b}">

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<annotation encoding="application/x-tex">{\displaystyle f(x)=ax+b}</annotation>

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</math></span><img src="./75c655df4be41082c4bba924beab2c1dc27d019c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.913ex; height:2.843ex;" alt="{\displaystyle f(x)=ax+b}" loading="lazy"></span> is defined by a linear polynomial in its argument, it is sometimes also referred to as being a "linear function", and the relationship between the argument and the function value may be referred to as a "linear relationship". This is potentially confusing, but usually the intended meaning will be clear from the context.

</p><p>The word <b>linear</b> comes from <a href="Latin" title="Latin">Latin</a> <i>linearis</i>, "pertaining to or resembling a line".

</p>

<div class="mw-heading mw-heading2"><h2 id="In_mathematics">In mathematics</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Linear_maps">Linear maps</h3></div>

<p>In mathematics, a <a href="Linear_map" title="Linear map">linear map</a> or <a href="Linear_function" title="Linear function">linear function</a> <i>f</i>(<i>x</i>) is a function that satisfies the two properties:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>

</p>

<ul><li><a href="Additive_map" title="Additive map">Additivity</a>: <span class="nowrap"><i>f</i>(<i>x</i> + <i>y</i>) = <i>f</i>(<i>x</i>) + <i>f</i>(<i>y</i>)</span>.</li>

<li><a href="Homogeneous_function" title="Homogeneous function">Homogeneity</a> of degree 1: <span class="nowrap"><i>f</i>(α<i>x</i>) = α <i>f</i>(<i>x</i>)</span> for all α.</li></ul>

<p>These properties are known as the <a href="Superposition_principle" title="Superposition principle">superposition principle</a>. In this definition, <i>x</i> is not necessarily a <a href="Real_number" title="Real number">real number</a>, but can in general be an <a href="Element_(mathematics)" title="Element (mathematics)">element</a> of any <a href="Vector_space" title="Vector space">vector space</a>. A more special definition of <a href="Linear_function#As_a_polynomial_function" title="Linear function">linear function</a>, not coinciding with the definition of linear map, is used in elementary mathematics (see below).

</p><p>Additivity alone implies homogeneity for <a href="Rational_number" title="Rational number">rational</a> α, since <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x+x)=f(x)+f(x)}">

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

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</math></span><img src="./10d3dd9de43b022b06f98fbd23c5988a5f16ac9c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.362ex; height:2.843ex;" alt="{\displaystyle f(x+x)=f(x)+f(x)}" loading="lazy"></span> implies <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(nx)=nf(x)}">

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

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</math></span><img src="./fe1988372667eb7701fdcdc77faab40926315db1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.723ex; height:2.843ex;" alt="{\displaystyle f(nx)=nf(x)}" loading="lazy"></span> for any <a href="Natural_number" title="Natural number">natural number</a> <i>n</i> by <a href="Mathematical_induction" title="Mathematical induction">mathematical induction</a>, and then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle nf(x)=f(nx)=f(m{\tfrac {n}{m}}x)=mf({\tfrac {n}{m}}x)}">

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</mfrac>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle nf(x)=f(nx)=f(m{\tfrac {n}{m}}x)=mf({\tfrac {n}{m}}x)}</annotation>

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</math></span><img src="./1aafc4698cd3399c0703d5e3a3f1954c9cd4db00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.394ex; height:3.009ex;" alt="{\displaystyle nf(x)=f(nx)=f(m{\tfrac {n}{m}}x)=mf({\tfrac {n}{m}}x)}" loading="lazy"></span> implies <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f({\tfrac {n}{m}}x)={\tfrac {n}{m}}f(x)}">

<mfrac>

</mfrac>

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<mfrac>

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<annotation encoding="application/x-tex">{\displaystyle f({\tfrac {n}{m}}x)={\tfrac {n}{m}}f(x)}</annotation>

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</math></span><img src="./869f6951f49015aeee8481360af71e154511dae1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.491ex; height:3.009ex;" alt="{\displaystyle f({\tfrac {n}{m}}x)={\tfrac {n}{m}}f(x)}" loading="lazy"></span>. The <a href="Dense_set" title="Dense set">density</a> of the rational numbers in the reals implies that any additive <a href="Continuous_function" title="Continuous function">continuous function</a> is homogeneous for any real number α, and is therefore linear.

</p><p>The concept of linearity can be extended to linear <a href="Operator_(mathematics)" title="Operator (mathematics)">operators</a>. Important examples of linear operators include the <a href="Derivative" title="Derivative">derivative</a> considered as a <a href="Differential_operator" title="Differential operator">differential operator</a>, and other operators constructed from it, such as <a href="Del" title="Del">del</a> and the <a href="Laplacian" class="mw-redirect" title="Laplacian">Laplacian</a>. When a <a href="Differential_equation" title="Differential equation">differential equation</a> can be expressed in linear form, it can generally be solved by breaking the equation up into smaller pieces, solving each of those pieces, and summing the solutions.

</p>

<div class="mw-heading mw-heading3"><h3 id="Linear_polynomials">Linear polynomials</h3></div>

<div role="note" class="hatnote navigation-not-searchable">Main articles: <a href="Linear_equation" title="Linear equation">Linear equation</a> and <a href="Linear_algebra" title="Linear algebra">Linear algebra</a></div>

<p>In a different usage to the above definition, a <a href="Polynomial" title="Polynomial">polynomial</a> of degree 1 is said to be linear, because the <a href="Graph_of_a_function" title="Graph of a function">graph of a function</a> of that form is a straight line.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>

</p><p>Over the reals, a simple example of a <a href="Linear_equation" title="Linear equation">linear equation</a> is given by

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<annotation encoding="application/x-tex">{\displaystyle y=mx+b,}</annotation>

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where <i>m</i> is often called the <a href="Slope" title="Slope">slope</a> or <a href="Gradient" title="Gradient">gradient</a>, and <i>b</i> the <a href="Y-intercept" title="Y-intercept"><i>y</i>-intercept</a>, which gives the point of intersection between the graph of the function and the <i>y</i> axis.

</p><p>Note that this usage of the term <i>linear</i> is not the same as in the section above, because linear polynomials over the real numbers do not in general satisfy either additivity or homogeneity. In fact, they do so <a href="If_and_only_if" title="If and only if">if and only if</a> the <a href="Constant_term" title="Constant term">constant term</a> – <i>b</i> in the example – equals 0. If <span class="nowrap"><i>b</i> ≠ 0</span>, the function is called an <b>affine function</b> (see in greater generality <a href="Affine_transformation" title="Affine transformation">affine transformation</a>).

</p><p><a href="Linear_algebra" title="Linear algebra">Linear algebra</a> is the branch of mathematics concerned with systems of linear equations.

</p>

<div class="mw-heading mw-heading3"><h3 id="Boolean_functions">Boolean functions</h3></div>

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

<p>In <a href="Boolean_algebra_(logic)" class="mw-redirect" title="Boolean algebra (logic)">Boolean algebra</a>, a linear function is a function <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">

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

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</math></span><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"></span> for which there exist <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{0},a_{1},\ldots ,a_{n}\in \{0,1\}}">

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<annotation encoding="application/x-tex">{\displaystyle a_{0},a_{1},\ldots ,a_{n}\in \{0,1\}}</annotation>

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</math></span><img src="./0f2ac49c0c13f2673abc93bcc3a9f6c5971a4c5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.753ex; height:2.843ex;" alt="{\displaystyle a_{0},a_{1},\ldots ,a_{n}\in \{0,1\}}" loading="lazy"></span> such that

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<annotation encoding="application/x-tex">{\displaystyle f(b_{1},\ldots ,b_{n})=a_{0}\oplus (a_{1}\land b_{1})\oplus \cdots \oplus (a_{n}\land b_{n})}</annotation>

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</math></span><img src="./602b2a363cad0729aaa1fe75ddb605cdb643bb13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.945ex; height:2.843ex;" alt="{\displaystyle f(b_{1},\ldots ,b_{n})=a_{0}\oplus (a_{1}\land b_{1})\oplus \cdots \oplus (a_{n}\land b_{n})}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1},\ldots ,b_{n}\in \{0,1\}.}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle b_{1},\ldots ,b_{n}\in \{0,1\}.}</annotation>

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</math></span><img src="./ba2ae645f396d30a8d80bf7f7611dbd95973c43e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.617ex; height:2.843ex;" alt="{\displaystyle b_{1},\ldots ,b_{n}\in \{0,1\}.}" loading="lazy"></span></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle a_{0}=1}</annotation>

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</math></span><img src="./e3873789cb6451e25f63b4d11572ac5c69d7873b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.545ex; height:2.509ex;" alt="{\displaystyle a_{0}=1}" loading="lazy"></span>, the above function is considered affine in linear algebra (i.e. not linear).

</p><p>A Boolean function is linear if one of the following holds for the function's <a href="Truth_table" title="Truth table">truth table</a>:

</p>

<ol><li>In every row in which the truth value of the function is <a href="Truth_value#Classical_logic" title="Truth value">T</a>, there are an odd number of Ts assigned to the arguments, and in every row in which the function is <a href="Truth_value#Classical_logic" title="Truth value">F</a> there is an even number of Ts assigned to arguments. Specifically, <span class="nowrap"><i>f</i>(F, F, ..., F) = F</span>, and these functions correspond to <a href="Linear_map" title="Linear map">linear maps</a> over the Boolean vector space.</li>

<li>In every row in which the value of the function is T, there is an even number of Ts assigned to the arguments of the function; and in every row in which the <a href="Truth_value" title="Truth value">truth value</a> of the function is F, there are an odd number of Ts assigned to arguments. In this case, <span class="nowrap"><i>f</i>(F, F, ..., F) = T</span>.</li></ol>

<p>Another way to express this is that each variable always makes a difference in the <a href="Truth_value" title="Truth value">truth value</a> of the operation or it never makes a difference.

</p><p><a href="Negation" title="Negation">Negation</a>, <a href="Logical_biconditional" title="Logical biconditional">Logical biconditional</a>, <a href="Exclusive_or" title="Exclusive or">exclusive or</a>, <a href="Tautology_(logic)" title="Tautology (logic)">tautology</a>, and <a href="Contradiction" title="Contradiction">contradiction</a> are linear functions.

</p>

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

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

<p>In <a href="Physics" title="Physics">physics</a>, <i>linearity</i> is a property of the <a href="Differential_equation" title="Differential equation">differential equations</a> governing many systems; for instance, the <a href="Maxwell_equations" class="mw-redirect" title="Maxwell equations">Maxwell equations</a> or the <a href="Diffusion_equation" title="Diffusion equation">diffusion equation</a>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>

</p><p>Linearity of a <a href="Homogeneous_differential_equation" title="Homogeneous differential equation">homogeneous differential equation</a> means that if two functions <i>f</i> and <i>g</i> are solutions of the equation, then any <a href="Linear_combination" title="Linear combination">linear combination</a> <span class="nowrap"><i>af</i> + <i>bg</i></span> is, too.

</p><p>In instrumentation, linearity means that a given change in an input variable gives the same change in the output of the measurement apparatus: this is highly desirable in scientific work. In general, instruments are close to linear over a certain range, and most useful within that range. In contrast, human senses are highly nonlinear: for instance, the brain completely ignores incoming light unless it exceeds a certain <a href="Absolute_threshold" title="Absolute threshold">absolute threshold</a> number of photons.

</p><p><a href="Linear_motion" title="Linear motion">Linear motion</a> traces a straight line trajectory.

</p>

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

<p>In <a href="Electronics" title="Electronics">electronics</a>, the linear operating region of a device, for example a <a href="Transistor" title="Transistor">transistor</a>, is where an output <a href="Dependent_variable" class="mw-redirect" title="Dependent variable">dependent variable</a> (such as the transistor collector <a href="Electric_current" title="Electric current">current</a>) is directly <a href="Proportionality_(mathematics)" title="Proportionality (mathematics)">proportional</a> to an input dependent variable (such as the base current). This ensures that an analog output is an accurate representation of an input, typically with higher amplitude (amplified). A typical example of linear equipment is a <a href="High_fidelity" title="High fidelity">high fidelity</a> <a href="Audio_amplifier" class="mw-redirect" title="Audio amplifier">audio amplifier</a>, which must amplify a signal without changing its waveform. Others are <a href="Linear_filter" title="Linear filter">linear filters</a>, and <a href="Linear_amplifier" title="Linear amplifier">linear amplifiers</a> in general.

</p><p>In most <a href="Science" title="Science">scientific</a> and <a href="Technology" title="Technology">technological</a>, as distinct from mathematical, applications, something may be described as linear if the characteristic is approximately but not exactly a straight line; and linearity may be valid only within a certain operating region—for example, a high-fidelity amplifier may distort a small signal, but sufficiently little to be acceptable (acceptable but imperfect linearity); and may distort very badly if the input exceeds a certain value.<sup id="cite_ref-Whitaker_4-0" class="reference"><a href="#cite_note-Whitaker-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>

</p>

<div class="mw-heading mw-heading3"><h3 id="Integral_linearity">Integral linearity</h3></div>

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

<p>For an electronic device (or other physical device) that converts a quantity to another quantity, Bertram S. Kolts writes:<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>

</p>

<blockquote><p>There are three basic definitions for integral linearity in common use: independent linearity, zero-based linearity, and terminal, or end-point, linearity. In each case, linearity defines how well the device's actual performance across a specified operating range approximates a straight line. Linearity is usually measured in terms of a deviation, or non-linearity, from an ideal straight line and it is typically expressed in terms of percent of <a href="Full_scale" title="Full scale">full scale</a>, or in ppm (parts per million) of full scale. Typically, the straight line is obtained by performing a least-squares fit of the data. The three definitions vary in the manner in which the straight line is positioned relative to the actual device's performance. Also, all three of these definitions ignore any gain, or offset errors that may be present in the actual device's performance characteristics.

<ul><li><a href="Linear_actuator" title="Linear actuator">Linear actuator</a></li>

<li><a href="Linear_element" class="mw-redirect" title="Linear element">Linear element</a></li>

<li><a href="Linear_foot" class="mw-redirect" title="Linear foot">Linear foot</a></li>

<li><a href="Linear_system" title="Linear system">Linear system</a></li>

<li><a href="Linear_programming" title="Linear programming">Linear programming</a></li>

<li><a href="Linear_differential_equation" title="Linear differential equation">Linear differential equation</a></li>

<li><a href="Bilinear_form" title="Bilinear form">Bilinear</a></li>

<li><a href="Multilinear_form" title="Multilinear form">Multilinear</a></li>

<li><a href="Linear_motor" title="Linear motor">Linear motor</a></li>

<li><a href="Linear_interpolation" title="Linear interpolation">Linear interpolation</a></li></ul>

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

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</style><cite id="CITEREFEdwards,_Harold_M.1995" class="citation book cs1">Edwards, Harold M. (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=ylFR4h5BIDEC&pg=PA78"><i>Linear Algebra</i></a>. Springer. p. 78. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780817637316</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="James_Stewart_(mathematician)" title="James Stewart (mathematician)">Stewart, James</a> (2008). <i>Calculus: Early Transcendentals</i>, 6th ed., Brooks Cole Cengage Learning. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-495-01166-8</bdi>, Section 1.2</span>

</li>

<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><cite id="CITEREFEvans2010" class="citation cs2"><a href="Lawrence_C._Evans" title="Lawrence C. Evans">Evans, Lawrence C.</a> (2010) [1998], <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/2000-37-03/S0273-0979-00-00868-5/S0273-0979-00-00868-5.pdf"><i>Partial differential equations</i></a> <span class="cs1-format">(PDF)</span>, <a href="Graduate_Studies_in_Mathematics" title="Graduate Studies in Mathematics">Graduate Studies in Mathematics</a>, vol. 19 (2nd ed.), Providence, R.I.: <a href="American_Mathematical_Society" title="American Mathematical Society">American Mathematical Society</a>, <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fgsm%2F019">10.1090/gsm/019</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-4974-3</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2597943">2597943</a>, <a rel="nofollow" class="external text" href="https://ghostarchive.org/archive/20221009/https://www.ams.org/journals/bull/2000-37-03/S0273-0979-00-00868-5/S0273-0979-00-00868-5.pdf">archived</a> <span class="cs1-format">(PDF)</span> from the original on 2022-10-09</cite></span>

</li>

<li id="cite_note-Whitaker-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-Whitaker_4-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFWhitaker2002" class="citation book cs1">Whitaker, Jerry C. (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=G5UHVIqEWdQC&pg=SA11-PA1"><i>The RF transmission systems handbook</i></a>. CRC Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8493-0973-1</bdi>.</cite></span>

</li>

<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><cite id="CITEREFKolts2005" class="citation web cs1">Kolts, Bertram S. (2005). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20120204065155/http://www.analogzone.com/nett1108.pdf">"Understanding Linearity and Monotonicity"</a> <span class="cs1-format">(PDF)</span>. analogZONE. Archived from <a rel="nofollow" class="external text" href="http://www.analogzone.com/nett1108.pdf">the original</a> <span class="cs1-format">(PDF)</span> on February 4, 2012<span class="reference-accessdate">. Retrieved <span class="nowrap">September 24,</span> 2014</span>.</cite></span>

</li>

<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><cite id="CITEREFKolts2005" class="citation journal cs1">Kolts, Bertram S. (2005). <a rel="nofollow" class="external text" href="http://caod.oriprobe.com/articles/9294129/Understanding_Linearity_and_Monotonicity.htm">"Understanding Linearity and Monotonicity"</a>. <i>Foreign Electronic Measurement Technology</i>. <b>24</b> (5): <span class="nowrap">30–</span>31<span class="reference-accessdate">. Retrieved <span class="nowrap">September 25,</span> 2014</span>.</cite></span>

</li>

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

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><span class="noviewer" typeof="mw:File"></span> The dictionary definition of <a href="https://en.wiktionary.org/wiki/Special:Search/linearity" class="extiw external" title="wiktionary:Special:Search/linearity"><i>linearity</i></a> at Wiktionary</li></ul></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-06-05" href="https://en.wikipedia.org/wiki/?title=Linearity&oldid=1294014917">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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