GL(n)

In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible. The group is so named because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.

To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over R (the set of real numbers) is the group of n×n invertible matrices of real numbers, and is denoted by GL_n(R) or GL(n, R).

More generally, the general linear group of degree n over any field F (such as the complex numbers), or a ring R (such as the ring of integers), is the set of n×n invertible matrices with entries from F (or R), again with matrix multiplication as the group operation. Typical notation is GL_n(F) or GL(n, F), or simply GL(n) if the field is understood.

More generally still, the general linear group of a vector space GL(V) is the abstract automorphism group, not necessarily written as matrices.

The special linear group, written SL(n, F) or SL_n(F), is the subgroup of GL(n, F) consisting of matrices with a determinant of 1.

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