Use Exercise 35 to show that if A and B are bases for a subspace W of R", then A cannot contain more vectors than B, and, conversely, B cannot contain more vectors than A.

Data from in Exercise 35

Suppose vectors b₁,...,b_p span a subspace W, and let {a₁,...,a_q} be any set in W containing more than p vectors. Fill in the details of the following argument to show that {a,...,aq} must be linearly dependent. First, let B = [b₁ ... b_p] and A = [a₁ ... a_q].