Definition:General Linear Group

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Definition

Let $K$ be a field.

The set of all nonsingular 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 $\map {\mathrm {GL}_n} K$ for the general linear group instead of $\GL {n, K}$.

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


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


Also see

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


Subgroups of the General Linear Group


Related Groups


Sources

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