Editing General linear group (section)

== General linear group of a vector space ==

If <math>V</math> is a [[vector space]] over the field <math>F</math>, the general linear group of <math>V</math>, written <math>\operatorname{GL}(V)</math> or <math>\operatorname{Aut}(V)</math>, is the group of all [[automorphism]]s of <math>V</math>, i.e. the set of all [[bijective]] [[linear transformation]]s <math>V\to V</math>, together with functional composition as group operation. If <math>V</math> has finite [[Hamel dimension|dimension]] <math>n</math>, then <math>\operatorname{GL}(V)</math> and <math>\operatorname{GL}(n,F)</math> are [[group isomorphism|isomorphic]]. The isomorphism is not canonical; it depends on a choice of [[basis (linear algebra)|basis]] in <math>V</math>. Given a basis <math>\{e_1,\dots,e_n\}</math> of <math>V</math> and an automorphism <math>T</math> in <math>\operatorname{GL}(V)</math>, we have then for every basis vector ''e''<sub>''i''</sub> that
: <math>T(e_i) = \sum_{j=1}^n a_{ji} e_j</math>
for some constants <math>a_{ij}</math> in <math>F</math>; the matrix corresponding to <math>T</math> is then just the matrix with entries given by the <math>a_{ji}</math>.

In a similar way, for a commutative ring <math>R</math> the group <math>\operatorname{GL}(n,R)</math> may be interpreted as the group of automorphisms of a ''[[free module|free]]'' <math>R</math>-module <math>M</math> of rank <math>n</math>. One can also define GL(''M'') for any <math>R</math>-module, but in general this is not isomorphic to <math>\operatorname{GL}(n,R)</math> (for any <math>n</math>).