- Identity function
mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument. In terms of equations, the function is given by "f"("x") = "x".
In others words, the function assigns to each element "x" of "M" the element "x" of "M".
The identity function "f" on "M" is often denoted by id"M" or 1"M".
In terms of
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 "M".
If "f" : "M" → "N" is any function, then we have "f" o id"M" = "f" = id"N" o "f" (where "o" denotes
function composition). In particular, id"M" is the identity elementof the monoidof 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 morphismin category theory, where the endomorphisms of "M" need not be functions.
*The identity function is a
linear operator, when applied to vector spaces.
*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 spacethe identity function is represented by the identity matrix"I""n", regardless of the basis.
metric spacethe identity is trivially an isometry. An object without any symmetryhas as symmetry groupthe trivial group only containing this isometry (symmetry type "C1).
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