An identity functionf is a function which doesn't have any effect: it always returns the same value that was used as its argument.
Formally, if M is a set, we define the identity function idM on M to be that function with domain and codomain M which satisfies
idM(x) = x for all elements x in M.
If f : M → N is any function, then we have f o idM = f = idN o f. In particular, idM is the identity element of the monoid of all functions from M to M.
Furthermore, in the case that V=W, this vector space is an associative algebra under composition of maps, since the composition of two linear maps is again a linear map, and composition of maps is always associative.
In the finite dimensional case and if bases have been chosen, then the composition of linear maps corresponds to the multiplication of matrices, the addition of linear maps corresponds to the addition of matrices, and the multiplication of linear maps with scalars corresponds to the multiplication of matrices with scalars.
The identity element of this algebra is the identitymap id : V → V.
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