Editing Linear map (section)

===Endomorphisms and automorphisms===
{{Main|Endomorphism|Automorphism}}
A linear transformation <math display="inline">f : V \to V</math> is an [[endomorphism]] of <math display="inline">V</math>; the set of all such endomorphisms <math display="inline">\operatorname{End}(V)</math>  together with addition, composition and scalar multiplication as defined above forms an [[associative algebra]] with identity element over the field <math display="inline">K</math> (and in particular a [[ring (algebra)|ring]]). The multiplicative identity element of this algebra is the [[identity function|identity map]] <math display="inline">\operatorname{id}: V \to V</math>.

An endomorphism of <math display="inline">V</math> that is also an [[isomorphism]] is called an [[automorphism]] of <math display="inline">V</math>. The composition of two automorphisms is again an automorphism, and the set of all automorphisms of <math display="inline">V</math> forms a [[group (math)|group]], the [[automorphism group]] of <math display="inline">V</math> which is denoted by <math display="inline">\operatorname{Aut}(V)</math> or <math display="inline">\operatorname{GL}(V)</math>. Since the automorphisms are precisely those [[endomorphisms]] which possess inverses under composition, <math display="inline">\operatorname{Aut}(V)</math> is the group of [[Unit (ring theory)|units]] in the ring <math display="inline">\operatorname{End}(V)</math>.

If <math display="inline">V</math> has finite dimension <math display="inline">n</math>, then <math display="inline"> \operatorname{End}(V)</math> is [[isomorphism|isomorphic]] to the [[associative algebra]] of all <math display="inline">n \times n</math> matrices with entries in <math display="inline">K</math>. The automorphism group of <math display="inline">V</math> is [[group isomorphism|isomorphic]] to the  [[general linear group]] <math display="inline">\operatorname{GL}(n, K)</math> of all <math display="inline">n \times n</math> invertible matrices with entries in <math display="inline">K</math>.