071 (13 1 1 2 3 Consider the matrix A = [ your work stepbystep. Correct answers not supported by substantive explanations will receive no credit.) . (For each of the question parts, briey justify your answer and show 1 point Find the LU factorization of A. (b) 1 point Solve A [2] = [g] for 322 using Cramer's Rule. ( ) ( ) (1 point) Find the eigenvalues of A. d(1 point) Explain why A is diagonalizable. (e) (1 point) Find an orthogonal basis W = {1712175} for R2 such that (71' = 171'. (f) (1 point) Divide each basis vector in W by its length to nd an orthonormal basis 8 = (13;7 1153}. (g) (1 point) Use the dot product to write the vector 6' = [1027] as a linear combination of the vectors in B. ( l ( ) 1 point Find F : A(AT) and show that it is symmetric poisitive denite. (h) 1 point If F is as in Part (h), write F as L1(LD7 where L1 is a lower triangular matrix. (bi-cm (22-55)]. 0 (275-55) (k) (1 point) If B = [131) 52)] and M is as in Part m show that A = BM. Note that M is triangular, and the columns of B are orthogonal vectors each with length 1. (j) (1 point) Find the matrix M: [