rank (of a matrix or linear map) Let A be an m × n matrix. The row rank and column rank of A are respectively the largest number of linearly independent rows or columns of A. Equivalently, they are the dimensions of the row space and column space, and the two ranks can be shown to be equal. This common value is the rank of A and also equals the determinantal rank or the number of non-zero rows when A is put into reduced echelon form. An n × n matrix is invertible if and only if it has rank n.
Given a linear map T:V→W between finite-dimensional vector spaces, the rank of T is the dimension of the image of T. If the matrix A represents T (see matrix of a linear map), then the rank of T equals the rank of A, as the image of T is now represented by the column space of A. See rank-nullity theorem. |