# Definition:General Linear Group

## Definition

Let $K$ be a field.

The set of all invertible order-$n$ square matrices over $K$ is a group under (conventional) matrix multiplication.

This group is called the general linear group (of degree $n$) and is denoted $\GL {n, K}$, or $\GL n$ if the field is implicit.

The field itself is usually $\R$, $\Q$ or $\C$, but can be any field.

### General Linear Group over Vector Space

Let $V$ be a vector space.

The group $\GL V$ is the group of all invertible linear transformations of $V$.

## Also denoted as

Some sources use the notation $\map {\operatorname {GL}_n} K$ instead of $\GL {n, K}$.

If $K$ is a Galois field of order $q$, the notations $\map {\operatorname {GL}_n} q$ and $\GL {n, q}$ are also seen.

Some sources use $\map {\operatorname {Gl} } {n, r}$ for $\GL {n, \R}$.

## Also see

• Results about the general linear group can be found here.