General Linear Group of A Vector Space
If V is a vector space over the field F, the general linear group of V, written GL(V) or Aut(V), is the group of all automorphisms of V, i.e. the set of all bijective linear transformations V → V, together with functional composition as group operation. If V has finite dimension n, then GL(V) and GL(n, F) are isomorphic. The isomorphism is not canonical; it depends on a choice of basis in V. Given a basis (e1, ..., en) of V and an automorphism T in GL(V), we have
for some constants ajk in F; the matrix corresponding to T is then just the matrix with entries given by the ajk.
In a similar way, for a commutative ring R the group GL(n, R) may be interpreted as the group of automorphisms of a free R-module M of rank n. One can also define GL(M) for any R-module, but in general this is not isomorphic to GL(n, R) (for any n).
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