# Transformation (function)

In mathematics, a transformation is a function f (usually with some geometrical underpinning) that maps a set X to itself, i.e. f : XX. In other areas of mathematics, a transformation may simply refer to any function, regardless of domain and codomain. For this wider sense of the term, see function (mathematics).

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.

More generally, a transformation in mathematics means a mathematical function (synonyms: "map" or "mapping"). A transformation can be an invertible function from a set X to itself, or from X to another set Y. The choice of the term transformation may simply indicate that the geometric aspects of a function are being considered (for example, with respect to invariants).

### 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: AB, 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.