C is column space

Let y be a vector in R". Let X = [X1, X2] where X has dimensions of n xp and X2 is nxq. Let M be the projection matrix for C(X) and let My be the projection matrix for C(Xi). X and X1 are not necessarily of full rank. a. Prove that MM = M1 and MM = M1. (Hint: C(M1) = C(Xi) C C(X) = C(M)) b. Let = My and i Miy. Prove that ||y i || = ||y ||? + 1|. Let y be a vector in R". Let X = [X1, X2] where X has dimensions of n xp and X2 is nxq. Let M be the projection matrix for C(X) and let My be the projection matrix for C(Xi). X and X1 are not necessarily of full rank. a. Prove that MM = M1 and MM = M1. (Hint: C(M1) = C(Xi) C C(X) = C(M)) b. Let = My and i Miy. Prove that ||y i || = ||y ||? + 1|