Use Exercise 27 to show that if A and B are bases for a subspace W of Rn, then A cannot contain more vectors than B, and, conversely, B cannot contain more vectors than A.
Exercise 27
Suppose vectors b1,,,,,,,,,bp span a subspace W, and let {a1,,,,,,,,,,aq} be any set in W containing more than p vectors. Fill in the details of the following argument to show that {a1,,,,,,,,,, aq} must be linearly dependent. First, let B = [b1 ...... bp] and A = [a1 .... aq].
a. Explain why for each vector aj, there exists a vector cj in Rp such that aj = Bcj
b. Let C = [c1 .... cq]. Explain why there is a nonzero vector u such that Cu = 0.
c. Use B and C to show that Au = 0. This shows that the columns of A are linearly dependent