general linear group

Given a vector spaceMathworldPlanetmath V, the general linear groupMathworldPlanetmath GL(V) is defined to be the group of invertible linear transformations from V to V. The group operationMathworldPlanetmath is defined by composition: given T:VV and T:VV in GL(V), the productPlanetmathPlanetmathPlanetmathPlanetmath TT is just the composition of the maps T and T.

If V=𝔽n for some field 𝔽, then the group GL(V) is often denoted GL(n,𝔽) or GLn(𝔽). In this case, if one identifies each linear transformation T:VV with its matrix with respect to the standard basis, the group GL(n,𝔽) becomes the group of invertiblePlanetmathPlanetmath n×n matrices with entries in 𝔽, under the group operation of matrix multiplicationMathworldPlanetmath.

One also discusses the general linear group on a module M over some ring R. There it is the set of automorphismsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of M as an R-module. For example, one might take GL(); this is isomorphic to the group of two-by-two matrices with integer entries having determinantMathworldPlanetmath ±1. If M is a general R-module, there need not be a natural interpretationMathworldPlanetmathPlanetmath of 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 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 spaceMathworldPlanetmath, GL(V) has a natural topology coming from the operator normMathworldPlanetmath; this is isomorphic to the topologyMathworldPlanetmath coming from its embeddingMathworldPlanetmath into the ring of matrices. When V is an infinite-dimensional vector space, some elements of GL(V) may not be continuousPlanetmathPlanetmath and one generally looks instead at the set of bounded operatorsMathworldPlanetmath.

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