Identity function
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In mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when Template:Mvar is the identity function, the equality Template:Math is true for all values of Template:Mvar to which Template:Mvar can be applied.
DefinitionEdit
Formally, if Template:Math is a set, the identity function Template:Math on Template:Math is defined to be a function with Template:Math as its domain and codomain, satisfying Template:Bi
In other words, the function value Template:Math in the codomain Template:Math is always the same as the input element Template:Math in the domain Template:Math. The identity function on Template:Mvar is clearly an injective function as well as a surjective function (its codomain is also its range), so it is bijective.<ref>Template:Cite book</ref>
The identity function Template:Math on Template:Math is often denoted by Template:Math.
In set theory, where a function is defined as a particular kind of binary relation, the identity function is given by the identity relation, or diagonal of Template:Math.<ref>Template:Cite book</ref>
Algebraic propertiesEdit
If Template:Math is any function, then Template:Math, where "∘" denotes function composition.<ref>Template:Cite book</ref> In particular, Template:Math is the identity element of the monoid of all functions from Template:Math to Template:Math (under function composition).
Since the identity element of a monoid is unique,<ref>Template:Cite book</ref> one can alternately define the identity function on Template:Math to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of Template:Math need not be functions.
PropertiesEdit
- The identity function is a linear operator when applied to vector spaces.<ref>Template:Citation</ref>
- In an Template:Mvar-dimensional vector space the identity function is represented by the identity matrix Template:Math, regardless of the basis chosen for the space.<ref>Template:Cite book</ref>
- The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.<ref>Template:Cite book</ref>
- In a metric space the identity function is trivially an isometry. An object without any symmetry has as its symmetry group the trivial group containing only this isometry (symmetry type Template:Math).<ref>Template:Aut, Hyperbolic Geometry, Springer 2005, Template:Isbn</ref>
- In a topological space, the identity function is always continuous.<ref>Template:Cite book</ref>
- The identity function is idempotent.<ref>Template:Cite book</ref>