Today I Learned

Some of the things I've learned every day since Oct 10, 2016

81: General Linear Groups

The general linear group \textrm{GL}(V), where V is a vector space, is the group of all automorphisms T: V \rightarrow V under the operation of composition of linear transformations.

When V is over the field F and is finite-dimensional with \textrm{dim}(V) = n, this group is isomorphic to the group of invertible n \times n matrices with entries from F under the operation of matrix multiplication. In this case, the group is often written as \textrm{GL}(n, F).

General linear groups are used in group representations. A group representation is a representation of a group as a general linear group. That is, it is an automorphism

f: G \rightarrow \textrm{GL}(V)

where G is the group being represented and V is any vector space.

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