which transforms a rectangular repetitive pattern
into a rhombic pattern. The four transformations are linear.
In mathematics, a transformation or self-map [1] is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. [2] [3] [4] 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. [5] [6]
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. [7]
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. [8]
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
-
Media related to
Transformation (function) at Wikimedia Commons
which transforms a rectangular repetitive pattern
into a rhombic pattern. The four transformations are linear.
In mathematics, a transformation or self-map [1] is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X. [2] [3] [4] 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. [5] [6]
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. [7]
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. [8]
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
-
Media related to
Transformation (function) at Wikimedia Commons