Note: N(A) denotes the null space of the matrix A, while R(A) stands for the range of A, viz., the column space of A. 1. (a) Let matrices A, B, C be such that A = BC. Show that R(A) C R(B), whereas N(C) C N(A). (b) Show that N(A"A) = N(A) for any matrix A, even rectangular. (Hint: Ax = 0 iff | Ax |= 0.) (c) Using (b), show that R(A"A) = R(AT). Hint 1. Let A, B be matrices such that R(A) C R(B). If rk(A) = rk(B), then show that the subspaces are equal. Hint 2. rank + nullity = dimension of codomain)