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 $\operatorname{GL} \left({n, K}\right)$, or $\operatorname{GL} \left({n}\right)$ if the field is implicit.

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

General Linear Group of Vector Space
Let $V$ be a vector space.

The group $\operatorname{GL}(V)$ is the group of all invertible linear transformations of $V$.

Also denoted as
Some sources use the notation $\operatorname{GL}_n \left({K}\right)$ instead of $\operatorname{GL} \left({n, K}\right)$.

If $K$ is a finite field of order $q$, the notations $\operatorname{GL}_n \left({q}\right)$ and $\operatorname{GL}\left({n,q}\right)$ are also seen.

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