PROBLEM 5 (10 pts.) Let A be n x n matrices with real entries (A ( Roxn). (1) (3 pts.) Assume that the matrix A is symmetric, i.e. A = A". Prove that Null( A) _ C(AT) where C(A" ) is the column space of AT. Hint: Consider ye C(AT), i.e. y = ATE for some Z. You must show that for any vector & belonging to Null(A), it holds that ya = 0. (2) (4 pts.) Assume that the matrix A is symmetric, i.e. A = A. We know that under this assumption all the eigenvalues of A belong to R (they are not complex). Prove carefully that any two eigenvectors of A (call them Ux and Up, both in the set R") corresponding to the distinct eigenvalues A and / respectively (X / /) are orthogonal Le. fu Hint: Starting from the definitions of the eigenvectors of A corresponding to the eigenvalues > and u respectively, you can easily prove that U. Xux = 0. Up by employing the fact that A = AT. Next use the fact that A #