In mathematics, a transformation, transform, or self-map is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. Examples include linear transformations of vector spaces and geometric transformations, which include projective transformations, affine transformations, and specific affine transformations, such as rotations, reflections and translations.
which transforms a rectangular repetitive pattern
into a rhombic pattern. The four transformations are linear.
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 subsets of some set X.
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 nn transformations and (n+1)n partial transformations.
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
- "Self-Map -- from Wolfram MathWorld". Retrieved March 4, 2024.
- Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. p. 1. ISBN 978-1-84800-281-4.
- Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 2. ISBN 978-0-8247-9662-4.
- Wilkinson, Leland (2005). The Grammar of Graphics (2nd ed.). Springer. p. 29. ISBN 978-0-387-24544-7.
- "Transformations". www.mathsisfun.com. Retrieved 2019-12-13.
- "Types of Transformations in Math". Basic-mathematics.com. Retrieved 2019-12-13.
- Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups. American Mathematical Society. p. 251. ISBN 978-1-4704-1493-1.
- Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction. Springer Science & Business Media. p. 2. ISBN 978-1-84800-281-4.
External links
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Media related to Transformation (function) at Wikimedia Commons
In mathematics a transformation transform or self map is a function f usually with some geometrical underpinning that maps a set X to itself i e f X X Examples include linear transformations of vector spaces and geometric transformations which include projective transformations affine transformations and specific affine transformations such as rotations reflections and translations A composition of four mappings coded in SVG which transforms a rectangular repetitive pattern into a rhombic pattern The four transformations are linear Partial transformationsWhile 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 subsets of some set X Algebraic structuresThe set of all transformations on a given base set together with function composition forms a regular semigroup CombinatoricsFor a finite set of cardinality n there are nn transformations and n 1 n partial transformations See alsoCoordinate transformation Data transformation statistics Geometric transformation Infinitesimal transformation Linear transformation List of transforms Rigid transformation Transformation geometry Transformation semigroup Transformation group Transformation matrixReferences Self Map from Wolfram MathWorld Retrieved March 4 2024 Olexandr Ganyushkin Volodymyr Mazorchuk 2008 Classical Finite Transformation Semigroups An Introduction Springer Science amp Business Media p 1 ISBN 978 1 84800 281 4 Pierre A Grillet 1995 Semigroups An Introduction to the Structure Theory CRC Press p 2 ISBN 978 0 8247 9662 4 Wilkinson Leland 2005 The Grammar of Graphics 2nd ed Springer p 29 ISBN 978 0 387 24544 7 Transformations www mathsisfun com Retrieved 2019 12 13 Types of Transformations in Math Basic mathematics com Retrieved 2019 12 13 Christopher Hollings 2014 Mathematics across the Iron Curtain A History of the Algebraic Theory of Semigroups American Mathematical Society p 251 ISBN 978 1 4704 1493 1 Olexandr Ganyushkin Volodymyr Mazorchuk 2008 Classical Finite Transformation Semigroups An Introduction Springer Science amp Business Media p 2 ISBN 978 1 84800 281 4 External linksMedia related to Transformation function at Wikimedia Commons
