View source for Transformation (function)

{{Short description|Function that applies a set to itself}}
{{Redirect|Transformation (mathematics)||Transformation (disambiguation)}}
{{broader|Function (mathematics)}}
[[File:A code snippet for a rhombic repetitive pattern.svg|thumb|upright=1.5|A [[Function composition|composition]] of four [[Map (mathematics)|mappings]] coded [[Scalable Vector Graphics|in&nbsp;SVG]],<br/>which '''transforms''' a [[Rectangle|rectangular]] repetitive [[pattern]]<br/>into a [[Rhombus|rhombic]] pattern. The four transformations are [[Linear map|linear]].]]
In [[mathematics]], a '''transformation''', '''transform''', or '''self-map'''<ref>{{Cite web|title=Self-Map -- from Wolfram MathWorld|url=https://mathworld.wolfram.com/Self-Map.html|access-date=March 4, 2024}}</ref> is a [[Function (mathematics)|function]] ''f'', usually with some [[Geometry|geometrical]] underpinning, that maps a [[set (mathematics)|set]] ''X'' to itself, i.e. {{nowrap|''f'': ''X'' → ''X''}}.<ref>{{cite book|author1=Olexandr Ganyushkin|author2=Volodymyr Mazorchuk|title=Classical Finite Transformation Semigroups: An Introduction|url=https://archive.org/details/classicalfinitet00gany_719|url-access=limited|year=2008|publisher=Springer Science & Business Media|isbn=978-1-84800-281-4|page=[https://archive.org/details/classicalfinitet00gany_719/page/n73 1]}}</ref><ref name="Grillet1995">{{cite book|author=Pierre A. Grillet|title=Semigroups: An Introduction to the Structure Theory|url=https://books.google.com/books?id=yM544W1N2UUC&pg=PA2|year=1995|publisher=CRC Press|isbn=978-0-8247-9662-4|page=2}}</ref><ref>{{cite book|author=Wilkinson, Leland |title=The Grammar of Graphics|publisher=Springer|year=2005|isbn=978-0-387-24544-7|page=29|url=https://books.google.com/books?id=NRyGnjeNKJIC&pg=PA29|edition=2nd}}</ref>
Examples include [[linear transformation]]s of [[vector spaces]] and [[geometric transformation]]s, which include [[projective transformation]]s, [[affine transformation]]s, and specific affine transformations, such as [[rotation]]s, [[reflection (mathematics)|reflections]] and [[translation (geometry)|translations]].<ref>{{Cite web|url=https://www.mathsisfun.com/geometry/transformations.html|title=Transformations|website=www.mathsisfun.com|access-date=2019-12-13}}</ref><ref name=":0">{{Cite web|url=https://www.basic-mathematics.com/transformations-in-math.html|title=Types of Transformations in Math|website=Basic-mathematics.com|access-date=2019-12-13}}</ref>
== Partial transformations ==
While it is common to use the term '''transformation''' for any function of a set into itself (especially in terms like "[[transformation semigroup]]" 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 [[partial functions]], then a '''partial transformation''' is a function ''f'': ''A'' → ''B'', where both ''A'' and ''B'' are [[subset]]s of some set ''X''.<ref name="Hollings2014">{{cite book|author=Christopher Hollings|title=Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups|url=https://books.google.com/books?id=O9wJBAAAQBAJ&pg=PA251|year=2014|publisher=American Mathematical Society|isbn=978-1-4704-1493-1|page=251}}</ref>
==Algebraic structures==
The set of all transformations on a given base set, together with [[function composition]], forms a [[regular semigroup]].
==Combinatorics==
For a finite set of [[cardinality]] ''n'', there are ''n''<sup>''n''</sup> transformations and (''n''+1)<sup>''n''</sup> partial transformations.<ref>{{cite book|author1=Olexandr Ganyushkin|author2=Volodymyr Mazorchuk|title=Classical Finite Transformation Semigroups: An Introduction|url=https://archive.org/details/classicalfinitet00gany_719|url-access=limited|year=2008|publisher=Springer Science & Business Media|isbn=978-1-84800-281-4|page=[https://archive.org/details/classicalfinitet00gany_719/page/n74 2]}}</ref>
==See also==
*[[Coordinate transformation]]
*[[Data transformation (statistics)]]
*[[Geometric transformation]]
*[[Infinitesimal transformation]]
*[[Linear transformation]]
*[[List of transforms]]
*[[Rigid transformation]]
*[[Transformation geometry]]
*[[Transformation semigroup]]
*[[Transformation group]]
*[[Transformation matrix]]
==References==
{{Reflist}}
==External links==
*{{Commonscatinline}}
{{Authority control}}
{{DEFAULTSORT:Transformation (Geometry)}}
[[Category:Transformation (function)| ]]
[[Category:Functions and mappings]]