Further Properties of Linear Equations

Abstract

We saw in chapter 2 that a set of vectors could be either linearly independent or linearly dependent, and in chapter 3 we considered vectors that were ‘nearly’ linearly dependent. We now wish to pursue these ideas a little further and ask whether or not it is reasonable to talk about the amount of linear independence of a set of vectors. To illustrate our meaning, consider two sets of nth order vectors, m vectors in each set, and let them form respectively the columns of two matrices A and B. Assume that there exists a vector x ≠ 0 such that Ax = 0, and that Ay = 0 only if y is a scalar multiple of x. Assume further that Bx = 0 and that, in addition, there exists a vector y ≠ 0 which is not a scalar multiple of x such that By = 0. Does this imply that, in some sense, the columns of A are more linearly independent than those of B? It is to this question and its implications that this chapter is principally devoted, and we first establish a result that gives a definitive meaning to the notion of the amount of linear independence.