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In `F33f`_`[mathematics`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Mathematics]`_`f, a `!transformation`!, `!transform`!, or `!self-map`!`:cite-ref-1[`F5bf`_`[1`#cite-note-1]`_`f] is a `F33f`_`[function`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_(mathematics)]`_`f `*f`*, usually with some `F33f`_`[geometrical`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Geometry]`_`f underpinning, that maps a `F33f`_`[set`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Set_(mathematics)]`_`f `*X`* to itself, i.e. `*f`*: `*X`* → `*X`*.`:cite-ref-2[`F5bf`_`[2`#cite-note-2]`_`f]`:cite-ref-grillet1995-3-0[`F5bf`_`[3`#cite-note-grillet1995-3]`_`f]`:cite-ref-4[`F5bf`_`[4`#cite-note-4]`_`f] Examples include `F33f`_`[linear transformations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_transformation]`_`f of `F33f`_`[vector spaces`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Vector_spaces]`_`f and `F33f`_`[geometric transformations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Geometric_transformation]`_`f, which include `F33f`_`[projective transformations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Projective_transformation]`_`f, `F33f`_`[affine transformations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Affine_transformation]`_`f, and specific affine transformations, such as `F33f`_`[rotations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rotation]`_`f, `F33f`_`[reflections`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Reflection_(mathematics)]`_`f and `F33f`_`[translations`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Translation_(geometry)]`_`f.`:cite-ref-5[`F5bf`_`[5`#cite-note-5]`_`f]`:cite-ref-0-6-0[`F5bf`_`[6`#cite-note-0-6]`_`f]

>>Contents

• `F0af`_`[Partial transformations`#partial-transformations]`_`f

• `F0af`_`[Algebraic structures`#algebraic-structures]`_`f

• `F0af`_`[Combinatorics`#combinatorics]`_`f

• `F0af`_`[See also`#see-also]`_`f

• `F0af`_`[References`#references]`_`f

• `F0af`_`[External links`#external-links]`_`f

-─

>>Partial transformations

While it is common to use the term `!transformation`! for any function of a set into itself (especially in terms like "`F33f`_`[transformation semigroup`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Transformation_semigroup]`_`f" 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 `F33f`_`[partial functions`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Partial_functions]`_`f, then a `!partial transformation`! is a function `*f`*: `*A`* → `*B`*, where both `*A`* and `*B`* are `F33f`_`[subsets`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Subset]`_`f of some set `*X`*.`:cite-ref-hollings2014-7-0[`F5bf`_`[7`#cite-note-hollings2014-7]`_`f]

>>Algebraic structures

The set of all transformations on a given base set, together with `F33f`_`[function composition`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Function_composition]`_`f, forms a `F33f`_`[regular semigroup`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Regular_semigroup]`_`f.

>>Combinatorics

For a finite set of `F33f`_`[cardinality`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Cardinality]`_`f `*n`*, there are `*n`*`*n`* transformations and (`*n`*+1)`*n`* partial transformations.`:cite-ref-8[`F5bf`_`[8`#cite-note-8]`_`f]

>>See also

• `F33f`_`[Endofunction`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Endofunction]`_`f

• `F33f`_`[Coordinate transformation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Coordinate_transformation]`_`f

• `F33f`_`[Data transformation (statistics)`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Data_transformation_(statistics)]`_`f

• `F33f`_`[Geometric transformation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Geometric_transformation]`_`f

• `F33f`_`[Infinitesimal transformation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Infinitesimal_transformation]`_`f

• `F33f`_`[Linear transformation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Linear_transformation]`_`f

• `F33f`_`[List of transforms`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=List_of_transforms]`_`f

• `F33f`_`[Rigid transformation`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Rigid_transformation]`_`f

• `F33f`_`[Transformation geometry`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Transformation_geometry]`_`f

• `F33f`_`[Transformation semigroup`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Transformation_semigroup]`_`f

• `F33f`_`[Transformation group`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Transformation_group]`_`f

• `F33f`_`[Transformation matrix`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=Transformation_matrix]`_`f

>>References

`:cite-note-1`!1.`! `F0af`_`[↑`#cite-ref-1]`_`f "Self-Map -- from Wolfram MathWorld". Retrieved March 4, 2024.

`:cite-note-2`!2.`! `F0af`_`[↑`#cite-ref-2]`_`f `:citerefolexandr-ganyushkinvolodymyr-mazorchuk2008`aOlexandr Ganyushkin; Volodymyr Mazorchuk (2008). `*Classical Finite Transformation Semigroups: An Introduction`*. Springer Science & Business Media. p. 1. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-1-84800-281-4.

`:cite-note-grillet1995-3`!3.`! `F0af`_`[↑`#cite-ref-grillet1995-3-0]`_`f `:citerefpierre-a-grillet1995`aPierre A. Grillet (1995). `*Semigroups: An Introduction to the Structure Theory`*. CRC Press. p. 2. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-8247-9662-4.

`:cite-note-4`!4.`! `F0af`_`[↑`#cite-ref-4]`_`f `:citerefwilkinson-leland2005`aWilkinson, Leland (2005). `*The Grammar of Graphics`* (2nd ed.). Springer. p. 29. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-0-387-24544-7.

`:cite-note-5`!5.`! `F0af`_`[↑`#cite-ref-5]`_`f "Transformations". `*www.mathsisfun.com`*. Retrieved 2019-12-13.

`:cite-note-0-6`!6.`! `F0af`_`[↑`#cite-ref-0-6-0]`_`f "Types of Transformations in Math". `*Basic-mathematics.com`*. Retrieved 2019-12-13.

`:cite-note-hollings2014-7`!7.`! `F0af`_`[↑`#cite-ref-hollings2014-7-0]`_`f `:citerefchristopher-hollings2014`aChristopher Hollings (2014). `*Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups`*. American Mathematical Society. p. 251. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-1-4704-1493-1.

`:cite-note-8`!8.`! `F0af`_`[↑`#cite-ref-8]`_`f `:citerefolexandr-ganyushkinvolodymyr-mazorchuk2008`aOlexandr Ganyushkin; Volodymyr Mazorchuk (2008). `*Classical Finite Transformation Semigroups: An Introduction`*. Springer Science & Business Media. p. 2. `F33f`_`[ISBN`:/page/wikibook/entry.mu`zim=wikipedia_en_all_nopic_2025-08.zim|entry_path=ISBN_(identifier)]`_`f 978-1-84800-281-4.

>>External links

• Media related to Transformation (function) at Wikimedia Commons

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