On the Rank of a Matrix

Abstract

Let A = (aij) be an n X m matrix with entries in a field F. Each of the n rows -of A can be regarded as a vector in coordinate m-space, Fm, and each of the m columns of A can be regarded as a vector in Fn. The row space of A is the subspace of Fin spanned by the rows of A and column space of A is the subspace of F71 spanned by the columns of A. The row rank of A is the dimension of the row space of A and the column rank of A is the dimension of the column space of A. It is proved in most undergraduate courses in linear algebra that the row rank and column rank are equal. This can be done by resorting to the determinants of square minors of A [3, pp. 22-24], or by computing the dimension of the solution space of a certain system of linear equations [2, pp. 47-51]. Products of linear transformations and the matrices associated with them can also be employed [1, pp. 234-235; 5, pp. 35, 42, 48-50; and 6, pp. 100-108]. In [4] Liebeck gives a short proof valid only for complex scalars. The purpose of this note is to present two very simple proofs that row rank and column rank are equal without resorting to any of these notions. Undergraduates should easily follow our arguments which the authors have not seen in any of the standard texts on linear algebra.