general linear group (original) (raw)

Given a vector space V, the general linear group 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⟶V and T′:V⟶V in GL⁡(V), the product T⁢T′ 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:V⟶V with its matrix with respect to the standard basis, the group GL⁡(n,𝔽) becomes the group of invertible n×n matrices with entries in 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 GL⁡(ℤ⊕ℤ); this is isomorphic to the group of two-by-two matrices with integer entries having determinant ±1. If M is a general R-module, there need not be a natural interpretation 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 space, 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 GL⁡(V) may not be continuous and one generally looks instead at the set of bounded operators.