Let w1,... wn be any basis of the subspace W Rm. Let A = (w1,...,wn) be

Question:

Let w1,... wn be any basis of the subspace W ⊂ Rm. Let A = (w1,...,wn) be the m x n matrix whose columns are the basis vectors, so that W = rng A and rank A = n. Let P = A(AT A)-1 AT be the corresponding projection matrix, as defined in Exercise 2.5.8.
(a) Prove that the orthogonal projection of v ∊ Rn onto w ∊ W is obtained by multiplying by the projection matrix: w = P v.
(b) Explain why Exercise 5.5.8 is a special case of this result.
(c) Show that if A = Q R is the factorization of Exercise 5.3.33, then P = Q QT. Why is P ≠ 1?