General Linear Groups

Why

We can generalize the real general linear groups to vector spaces over $\C $.

Definition

Suppose $V$ is a vector space over the field $\C $ of complex numbers. The set of isomorphisms of $V$ onto itself is a group, called the general linear group, under the operation of composition. If $V$ has dimension $n$, then the general linear group can be identified with the invertible $n \times n$ complex matrices in the usual way.

Notation

We denote by $GL(V)$ the general linear group of isomorphisms of $V$ onto itself. If $f \in GL(V)$, and $V$ has a finite basis $e_1, \dots , e_n \in V$, then $f$ has corresponding matrix representation $A \in \C ^{n \times n}$ given by

\[ A = \bmat{ f(e_1) & \cdots & f(e_n)}. \]