THEOREM. If A is an m x n matrix, then the row rank of A is equal to the column rank of A.

Proof. If A = 0, then the row and column rank of A are both 0; otherwise, let r be the smallest positive integer such that there is an m x r matrix B and an r x n matrix C satisfying A = BC. Thus the r rows of C form a minimal spanning set of the row space of A and the r columns of B form a minimal spanning set of the column space of A. Hence, row and column ranks are both r. #