Problem 6.1. Let A be an m x n matrix, and suppose the CR factorization of A is A = CR, where C is an m x d matrix composed of linearly independent columns of A, and R is a d x n matrix in RREF without any all-zeros rows. For each of the following statements, determine whether it is true or false. If you answer "true", explain your reasoning. If you answer "false", give a counterexample. (a) CS(A) = CS(R). (c) CS(AT) = CS(RT ). (b) dim(CS(A)) = dim(CS(R)). (d) NS(A) = NS(R).Problem 6.4. Prove that if an m x 72. matrix A is not square (i.e., m 7 n), then the columns of at least one of A and AT are linearly dependent. Problem 6.5. Prove that if an m X n matrix A has rank 1, then there exist an mvector u and an nvector v such that A = uvT