Rank, Inner Product and Nonsingularity

Abstract

Let A be an m × n matrix. The subspace of R m spanned by the column vectors of A is called the column space or the column span of A and is denoted by C (A). Similarly the subspace of R n spanned by the row vectors of A is called the row space of A, denoted by R(A). Clearly R(A) is isomorphic to C (A). The dimension of the column space is called the column rank whereas the dimension of the row space is called the row rank of the matrix. These two definitions turn out to be very short-lived in any linear algebra book since the two ranks are always equal as we show in the next result. 2.1 The column rank of a matrix equals its row rank. Proof Let A be an m × n matrix with column rank r. Then C (A) has a basis of r vectors, say b 1 , . . . , b r . Let B be the m × r matrix [b 1 , . . . , b r ]. Since every column of A is a linear combination of b 1 , . . . , b r , we can write A = BC for some r × n matrix C. Then every row of A is a linear combination of the rows of C and therefore R(A) ⊂ R(C). It follows by 1.7 that the dimension of R(A), which is the row rank of A, is at most r. We can similarly show that the column rank does not exceed the row rank and therefore the two must be equal.