Let A be an m x n matrix with rank A = n. (a) Prove that the

Question:

Let A be an m x n matrix with rank A = n.
(a) Prove that the matrix P = A(ATA)~l AT is a projection matrix, meaning that P2 = P, cf. Exercise 2.5.8.
(b) Construct the projection matrix corresponding to
(i)

(ii)

(iii)

(iv)

(c) Prove that P is symmetric.
(d) Prove that mg P = mg A.
(e) Show that v* = P b is the closest point on the subspace mg A - rng P to b.
(f) Show that if A is nonsingular, then P = I. How do you interpret this in light of part (e)?