Given a vector space , the general linear group is defined to be the group of invertible linear transformations from to . The group operation is defined by composition: given and in , the product is just the composition of the maps and .

If for some field , then the group is often denoted or . In this case, if one identifies each linear transformation with its matrix with respect to the standard basis, the group becomes the group of invertible matrices with entries in , under the group operation of matrix multiplication.

One also discusses the general linear group on a module over some ring . There it is the set of automorphisms of as an -module. For example, one might take ; this is isomorphic to the group of two-by-two matrices with integer entries having determinant . If is a general -module, there need not be a natural interpretation of 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 for different vector spaces and different fields . The general linear group is an algebraic group, and it is a Lie group if is a real or complex vector space.

When is a finite-dimensional Banach space, 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 is an infinite-dimensional vector space, some elements of may not be continuous and one generally looks instead at the set of bounded operators.