general linear group

Given a vector space $V$ , the general linear group ${\operatorname{GL}}(V)$ is defined to be the group of invertible linear transformations from $V$ to $V$ . The group operation is defined by composition: given $T:V\longrightarrow V$ and $T^{\prime}:V\longrightarrow V$ in ${\operatorname{GL}}(V)$ , the product $TT^{\prime}$ is just the composition of the maps $T$ and $T^{\prime}$ .

If $V=\mathbb{F}^{n}$ for some field $\mathbb{F}$ , then the group ${\operatorname{GL}}(V)$ is often denoted ${\operatorname{GL}}(n,\mathbb{F})$ or ${\operatorname{GL}}_{n}(\mathbb{F})$ . In this case, if one identifies each linear transformation $T:V\longrightarrow V$ with its matrix with respect to the standard basis, the group ${\operatorname{GL}}(n,\mathbb{F})$ becomes the group of invertible $n\times n$ matrices with entries in $\mathbb{F}$ , under the group operation of matrix multiplication.

One also discusses the general linear group on a module $M$ over some ring $R$ . There it is the set of automorphisms of $M$ as an $R$ -module. For example, one might take ${\operatorname{GL}}(\mathbb{Z}\oplus\mathbb{Z})$ ; this is isomorphic to the group of two-by-two matrices with integer entries having determinant $\pm 1$ . If $M$ is a general $R$ -module, there need not be a natural interpretation of ${\operatorname{GL}}(M)$ as a matrix group.

The general linear group is an example of a group scheme; viewing it in this way ties together the properties of ${\operatorname{GL}}(V)$ for different vector spaces $V$ and different fields $F$ . The general linear group is an algebraic group, and it is a Lie group if $V$ is a real or complex vector space.

When $V$ is a finite-dimensional Banach space, ${\operatorname{GL}}(V)$ has a natural topology coming from the operator norm; this is isomorphic to the topology coming from its embedding into the ring of matrices. When $V$ is an infinite-dimensional vector space, some elements of ${\operatorname{GL}}(V)$ may not be continuous and one generally looks instead at the set of bounded operators.

Title	general linear group
Canonical name	GeneralLinearGroup
Date of creation	2013-03-22 12:25:36
Last modified on	2013-03-22 12:25:36
Owner	djao (24)
Last modified by	djao (24)
Numerical id	8
Author	djao (24)
Entry type	Definition
Classification	msc 20G15
Related topic	Group
Related topic	Representation
Related topic	SpecialLinearGroup