NAME: Professor/Section: For each problem show ALL your steps and clearly BOX your final answer. Problem Points Possible Points Earned 1 10 2 10 3 15 4 5 5 10 6 10 7 10 8 15 9 10 10 5 Total 100 1 1. Let A = 4 0 1 −2 1 0 −2 0 1 . a) Find a basis for the column space of A, and state dim col A. b) Find a basis for the null space of A, and state dim nul A. c) Find a basis for the row space of A, and state dim row A. d) Find a vector perpendicular to row A. 2 skip this page 2. Prove that, for an n×n matrix, A, row A = (nul A)⊥. Hints: first show that if x is in row A, then x is orthogonal to every u in nul A - what does this show? Then assume rank A = r and examine the dimensions of nul A and (nul A)⊥ and finish the proof. 4 skip this page 3. A = bracketleftbigg1 −3 2 −4 bracketrightbigg a) Find the determinant of A. b) Find the eigenvalues and eigenvectors of A. c) Is A diagonalizable? If yes,justify your answer and then find an invertable matrix P that diagonalizes A and check that it is correct. If no, justify your answer. d) Is it possible to find an orthogonal matrix U which diagonalizes A? Justify your answer. 6 skip this page 4. Let a matrix A have a characteristic polynomial λ(λ + 5)(λ− 2)2. Is A invertible? Justifly your answer. 8 skip this page 5. Consider the linearly independent set of vectors S = 1 2 0 3 , 4 0 5 8 , 8 1 5 6 , that define a subspace of R4, such that W = span{S}. (a) Find an orthonormal basis, ˆS for W. (b) Find a vector g so that the four vectors ˆS and g form an orthonor- mal basis for R4. 10 skip this page 6. Let A and B be n×n matrices that commute with each other (A B = B A). Prove that Ak B = B Ak. 12 skip this page 7. a) Orthogonally diagonalize A = bracketleftbigg 1 −2 −2 1 bracketrightbigg . b) Let B = bracketleftbigg5 −5 1 1 bracketrightbigg , find an invertible matrix P and a matrix C of the form bracketleftbigga −b b s bracketrightbigg such that B = P C P−1. 14 skip this page 8. Which of the following define an inner product on their respective vec- tor space (justify your answer): a) 〈x,y〉 = x21 y1 + x22 y2 x, y ∈ R4 b) 〈x,y〉 = x1 y1 + x2 y2 x, y ∈ R3 c) 〈x,y〉 = x1 y3 + x2 y2 + x3 y1 x, y ∈ R3 d) 〈x,y〉 = 2 x21 + 2 x22 −y21 −y22 x, y ∈ R3 e) 〈x,y〉 = x1 y1 −x2 y2 −x3 y3 x, y ∈ R3 16 skip this page 9. Prove that (AB)T = BTAT. 18 skip this page 10. Let W = span 1 2 1 1 , 0 1 0 1 . Find the least squares approximation of b = 1 3 2 1 in W. 20 skip this page