Identity function
Encyclopedia : I : ID : IDE : Identity function
In mathematics, an identity function, also called identity map or identity transformation, is a function which does not have any effect: it always returns the same value that was used as its argument. In other words, the identity function is the function f(x) = x.
Definition
Formally, if M is a set, the identity function f on M is defined to be that function with domain and codomain M which satisfies- f(x) = x for all elements x in M.
Algebraic property
If f : M → N is any function, then we have f o idM = f = idN o f (where "o" denotes function composition). In particular, idM is the identity element of the monoid of all functions from M to M.Since the identity element of a monoid is unique, one can alternately define the identity function on M to be this identity element. Such a definition generalizes to the concept of an identity morphism in category theory, where the endomorphisms of M need not be functions.
Examples
- The identity function on the positive integers is a completely multiplicative function (essentially multiplication by 1), considered in number theory.
- In an n-dimensional vector space the identity function is represented by the identity matrix In, regardless of the basis.
- In a metric space the identity is trivially an isometry. An object without any symmetry has as symmetry group the trivial group only containing this isometry (symmetry type ''C1).
See also
From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.