Let Y be response, X be n xp design matrix with rank(X) =p (so that (XX)-¹ exists). Let H = X(XX)-¹XT be nxn hat matrix associated with X. (a) Show that tr(H) = p and H2 = H. (10pts) (b) Suppose that X can be partitioned as X = [X1, X2], where X₁ is nx P1, X2 is n x P2, and p₁ + P2 = p. Moreover, suppose that every column in X, is orthogonal to X2, i.e., XX2 = 0. Show that H- H₁ + H2, where (c) (b1, H₁ =X₁(XX₁)¯¹XT H₂ = X2(XX2) 'X == For any n x 1 vector a = = (a1,.,an) and b = bn) (as and b;s are scalars), if ab = 0, then ||ab||2 ||a||||b|| (This is a famous Pythagorean Theorem), where ||a||2: a. Under conditions in (b), show that ||Ŷ ||² = ||Ŷ ₁||2 + ||Ŷ 2||2 where ŶHY, Ŷ₁ = H₁Y₁ and Ŷ2 = H2Y2. (10pts) -