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.

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

 * General Linear Group is Group

Subgroups of the General Linear Group

 * Definition:Special Linear Group
 * Definition:Unitary Group
 * Definition:Special Unitary Group
 * Definition:Orthogonal Group
 * Definition:Symplectic Group
 * Definition:Triangular Matrix Group

Related Groups

 * Definition:Projective Linear Group
 * Definition:Affine Group
 * Definition:General Semilinear Group
 * Definition:Infinite General Linear Group