Problem 2.2 Assume p S n and let X E 13\"\" be a full-rank matrixl, and H = X(XTX)1XT (1) Note that H is a square n x n, matrix. This problem is devoted to understanding the properties H. (a) Show that H is symmetric (H T = H) and idempotent (H2 = H). Hm: (AB)T = BTAT. Use this to same (ABC)T = CTBTAT. Also (A_1)T = (AT)-l, hence an invertible matrix is symmetric if and only if its nner'se is symmetc. 1This together with p 5 it guarantees that X TX is invertible, which makes H weH-dened. Any matrix that satises conditions in (a) is an orthogonal projection mam. In this problem, we will verify this directly for the H given in (1). Let V = Im(X). (b) Show that for any y E El\