Identity function

An identical picture or identity is in mathematics, a function that returns exactly their argument. Although the identity mapping is often abbreviated by " identity ", they should not be confused with an identity equation.

Definition

Be a lot, then the identity map on defined by:

That is true for each of

The identity mapping is thus a bijection, its graph is the set

The index is often omitted when the context is clear, the definition set. In this case, it is also written instead. Instead of the notation sometimes the notation used.

Properties

Is an arbitrary function, then for the composition ( sequential execution ) with the identity:

And

Therefore, in the amount of all functions from to the identity of the neutral element with respect to the composition. Thus, these functions form a monoid. In particular, the identity is the identity element in the group of the permutations of the set.

The identity of the set of natural numbers is a multiplicative function, considered in the number theory.

On a topological space, the identity is a continuous function. On a topological vector space, for example a Banach space, the identity is a continuous linear operator, the operator is referred to as one. If the finite-dimensional Banach space additionally, the identity is compact.

The matrix multiplication with the unit matrix ( identity element ) is an identity mapping. In linear algebra Basiswechselmatrizen can be construed as representing matrices of the identity map with respect to two different bases.

Itemization

• Mathematical concept
• Mathematical function