Problem 1 Consider the linear model y = XB toe where y is a n x 1 vector of observations, X is the n x p design matrix, B is the p x 1 vector of coefficients, o' is the variance and e is a n x 1 vector of independent standard normal random variables. (1) (8 pts) Using the notations in slides, QX'XQ' = A2 PXQ' = D = where A = diag()1, 12, ..., Ap) is the diagonal matrix with entries the square roots of the eigenvalues of X'X, do a Principal Component Analysis (PCA) using the linear transformation T = XQ'. Set y = QB, find XS in terms of T and y. (2) (10 pts) Hence, show that regressing y on T is equivalent to regressing z = Py on D