Given multivariate data X₁,..., X_N (in R^P) in mean- deviation form, let P be a p x p matrix, and define Y_k = P^T X_k for k = 1,..., N. 

a. Show that Y₁,..., Y_N are in mean-deviation form. Let w be the vector in R^N with a 1 in each entry. Then [X₁ .......... X_N]w = 0 (the zero vector in R^P.

b. Show that if the covariance matrix of X₁,..., X_N is S, then the covariance matrix of Y₁,..., Y_N is P^TSP.