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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Transformation (function)</span></span>
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</style><div role="note" class="hatnote navigation-not-searchable">"Transformation (mathematics)" redirects here. For other uses, see <a href="Transformation_(disambiguation)" class="mw-redirect mw-disambig" title="Transformation (disambiguation)">Transformation (disambiguation)</a>.</div>
<div role="note" class="hatnote navigation-not-searchable">For broader coverage of this topic, see <a href="Function_(mathematics)" title="Function (mathematics)">Function (mathematics)</a>.</div>
<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>transformation</b>, <b>transform</b>, or <b>self-map</b><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> is a <a href="Function_(mathematics)" title="Function (mathematics)">function</a> <i>f</i>, usually with some <a href="Geometry" title="Geometry">geometrical</a> underpinning, that maps a <a href="Set_(mathematics)" title="Set (mathematics)">set</a> <i>X</i> to itself, i.e. <span class="nowrap"><i>f</i>: <i>X</i> → <i>X</i></span>.<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><sup id="cite_ref-Grillet1995_3-0" class="reference"><a href="#cite_note-Grillet1995-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup>
Examples include <a href="Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformations</a> of <a href="Vector_spaces" class="mw-redirect" title="Vector spaces">vector spaces</a> and <a href="Geometric_transformation" title="Geometric transformation">geometric transformations</a>, which include <a href="Projective_transformation" class="mw-redirect" title="Projective transformation">projective transformations</a>, <a href="Affine_transformation" title="Affine transformation">affine transformations</a>, and specific affine transformations, such as <a href="Rotation" title="Rotation">rotations</a>, <a href="Reflection_(mathematics)" title="Reflection (mathematics)">reflections</a> and <a href="Translation_(geometry)" title="Translation (geometry)">translations</a>.<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-:0_6-0" class="reference"><a href="#cite_note-:0-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="Partial_transformations">Partial transformations</h2></div>
<p>While it is common to use the term <b>transformation</b> for any function of a set into itself (especially in terms like "<a href="Transformation_semigroup" title="Transformation semigroup">transformation semigroup</a>" and similar), there exists an alternative form of terminological convention in which the term "transformation" is reserved only for bijections. When such a narrow notion of transformation is generalized to <a href="Partial_functions" class="mw-redirect" title="Partial functions">partial functions</a>, then a <b>partial transformation</b> is a function <i>f</i>: <i>A</i> → <i>B</i>, where both <i>A</i> and <i>B</i> are <a href="Subset" title="Subset">subsets</a> of some set <i>X</i>.<sup id="cite_ref-Hollings2014_7-0" class="reference"><a href="#cite_note-Hollings2014-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="Algebraic_structures">Algebraic structures</h2></div>
<p>The set of all transformations on a given base set, together with <a href="Function_composition" title="Function composition">function composition</a>, forms a <a href="Regular_semigroup" title="Regular semigroup">regular semigroup</a>.
</p>
<div class="mw-heading mw-heading2"><h2 id="Combinatorics">Combinatorics</h2></div>
<p>For a finite set of <a href="Cardinality" title="Cardinality">cardinality</a> <i>n</i>, there are <i>n</i><sup><i>n</i></sup> transformations and (<i>n</i>+1)<sup><i>n</i></sup> partial transformations.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Endofunction" class="mw-redirect" title="Endofunction">Endofunction</a></li>
<li><a href="Coordinate_transformation" class="mw-redirect" title="Coordinate transformation">Coordinate transformation</a></li>
<li><a href="Data_transformation_(statistics)" title="Data transformation (statistics)">Data transformation (statistics)</a></li>
<li><a href="Geometric_transformation" title="Geometric transformation">Geometric transformation</a></li>
<li><a href="Infinitesimal_transformation" title="Infinitesimal transformation">Infinitesimal transformation</a></li>
<li><a href="Linear_transformation" class="mw-redirect" title="Linear transformation">Linear transformation</a></li>
<li><a href="List_of_transforms" title="List of transforms">List of transforms</a></li>
<li><a href="Rigid_transformation" title="Rigid transformation">Rigid transformation</a></li>
<li><a href="Transformation_geometry" title="Transformation geometry">Transformation geometry</a></li>
<li><a href="Transformation_semigroup" title="Transformation semigroup">Transformation semigroup</a></li>
<li><a href="Transformation_group" class="mw-redirect" title="Transformation group">Transformation group</a></li>
<li><a href="Transformation_matrix" title="Transformation matrix">Transformation matrix</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/Self-Map.html">"Self-Map -- from Wolfram MathWorld"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">March 4,</span> 2024</span>.</cite></span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFOlexandr_GanyushkinVolodymyr_Mazorchuk2008" class="citation book cs1">Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/classicalfinitet00gany_719"><i>Classical Finite Transformation Semigroups: An Introduction</i></a></span>. Springer Science & Business Media. p. <a rel="nofollow" class="external text" href="https://archive.org/details/classicalfinitet00gany_719/page/n73">1</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-84800-281-4</bdi>.</cite></span>
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<li id="cite_note-Grillet1995-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-Grillet1995_3-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFPierre_A._Grillet1995" class="citation book cs1">Pierre A. Grillet (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=yM544W1N2UUC&pg=PA2"><i>Semigroups: An Introduction to the Structure Theory</i></a>. CRC Press. p. 2. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8247-9662-4</bdi>.</cite></span>
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</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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mathsisfun.com/geometry/transformations.html">"Transformations"</a>. <i>www.mathsisfun.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-13</span></span>.</cite></span>
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<li id="cite_note-:0-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-:0_6-0">^</a></b></span> <span class="reference-text"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.basic-mathematics.com/transformations-in-math.html">"Types of Transformations in Math"</a>. <i>Basic-mathematics.com</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-13</span></span>.</cite></span>
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<li id="cite_note-Hollings2014-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-Hollings2014_7-0">^</a></b></span> <span class="reference-text"><cite id="CITEREFChristopher_Hollings2014" class="citation book cs1">Christopher Hollings (2014). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=O9wJBAAAQBAJ&pg=PA251"><i>Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups</i></a>. American Mathematical Society. p. 251. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4704-1493-1</bdi>.</cite></span>
</li>
<li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text"><cite id="CITEREFOlexandr_GanyushkinVolodymyr_Mazorchuk2008" class="citation book cs1">Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). <span class="id-lock-limited" title="Free access subject to limited trial, subscription normally required"><a rel="nofollow" class="external text" href="https://archive.org/details/classicalfinitet00gany_719"><i>Classical Finite Transformation Semigroups: An Introduction</i></a></span>. Springer Science & Business Media. p. <a rel="nofollow" class="external text" href="https://archive.org/details/classicalfinitet00gany_719/page/n74">2</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-84800-281-4</bdi>.</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> Media related to <a href="https://commons.wikimedia.org/wiki/Category:Transformation_(function)" class="extiw external" title="commons:Category:Transformation (function)">Transformation (function)</a> at Wikimedia Commons</li></ul>
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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/Q12202238#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1121" 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/Q12202238#identifiers&#124;class=noprint&#124;Edit_this_at_Wikidata1121" 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><span class="uid"><a rel="nofollow" class="external text" href="https://id.loc.gov/authorities/sh85136920">United States</a></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Transformations (mathématiques)"><a rel="nofollow" class="external text" href="https://catalogue.bnf.fr/ark:/12148/cb11977766v">France</a></span></span></li><li><span class="uid"><span class="rt-commentedText tooltip tooltip-dotted" title="Transformations (mathématiques)"><a rel="nofollow" class="external text" href="https://data.bnf.fr/ark:/12148/cb11977766v">BnF data</a></span></span></li><li><span class="uid"><a rel="nofollow" class="external text" href="https://www.nli.org.il/en/authorities/987007546196405171">Israel</a></span></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><span class="uid"><a rel="nofollow" class="external text" href="https://lux.collections.yale.edu/view/concept/8294d20c-6018-4a98-b21c-fda34af225ca">Yale LUX</a></span></li></ul></div></td></tr></tbody></table></div></div><!--htdig_noindex--><div><div class="zim-footer">
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