# 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.