Problem 8.1 (6 points). Let A E Rnxm. The Singular Value Decomposition (SVD) tells us that A = UEVT where U E Rnxn and V E Rmam are orthogonal matrices, and E E Rnxm satisfies Ell > >22 2 . . . 2 0 and Eij =0 for i # j. Recall that the columns u1, . .., Un of U are called the left singular vectors of A, the columns v1, ..., Um of V are called the right singular vectors of A, and the diagonal entries oi = Eni are called the singular values of A. Denote the rank of A by r, and recall from lecture that r = # {i | Zai * 0}. (a) Let U u1 ur ERnxr = V1 ERmxr and let E E R"XT be the diagonal matrix with entries Eni = oi. Show that A = UDVT. (Hint: show that the ijth entry of UEVT and UEVT match for all i, j.) (b) Give an orthonormal basis of Ker(A) in terms of the singular vectors of A. Justify. (c) Give an orthonormal basis of Im(A) in terms of the singular vectors of A. Justify