General Linear Group

The general linear group of degree n is the set of n by n invertible matrices together with the operation of ordinary matrix multiplication, which forms a group because the product of two invertible matrices is again invertible and every invertible matrix has an inverse, with the identity matrix serving as the group's identity element. The group is so named because the columns, and equally the rows, of an invertible matrix are linearly independent, so matrices in the general linear group carry points in general linear position to other points in general linear position. Written GLn(F) for matrix entries drawn from a field F, or more generally over a ring, the group's subgroup of matrices with determinant equal to one is called the special linear group. General linear groups are central to the theory of group representations and arise throughout the study of spatial symmetries, symmetries of vector spaces, and polynomials; for degree two or higher the group is not abelian.