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NAME Id. No, Second Midterm Examination, Math 304 Fall 2012, Wilkerson Section. November 29, 2012 – 75 minutes No notes, books, calculators, music players, earphones, etc. Show all work. Problem Score I.(20) II.(15) III.(20) IV.(20) V.(12) VI.(13) Total(100) 1. 20 points (15 points) Let 0 2 A= −1 3 a) Give the charactaristic polynomial of A and its eigenvalues. (8 points) b) Give a (nonzero) eigenvector for each eigenvalue. (12 points). 1 2. 15 points The matrix 3 2 −2 A = 1 3 −1 1 2 0 has eigenvalues 1, 2, and 3. Find a nonzero eigenvector for each eigenvalue. 2 3. 20 points Let 1 1 2 A = 1 3 , ~b = 2 . 5 1 4 Let v1 and v2 be the columns of A, and P its column space. a) Calculate the projection of v2 onto v1 , projv1 (v2 ). (7 points) b) Use this to calculate an orthonormal basis of P . (7 points). 1 c) Use this basis to calculate projP (b). for b = 2. 5 3 4. 20 points Let 1 1 0 ~ A = 0 1 , b = 2 . 5 1 1 Let P be the column space of A. a) Calculate the least squares solution y to Ax = b. 10 points b) Calculate the projection of b onto P , projP (b). 5 points c) Calculate the distance from b to P . 5 points 4 5. Let L : R3 → R2 be the linear transformation with L(1, 1, 0)T = (1, 0)T , L(1, 0, 1)T = (1, 2)T , and L(1, 1, 1)T = (2, 1)T . Find a 2x3 matrix A such that Ax = Lx. Hint: find L(1, 0, 0)T , etc. 12 points 6. (13 points) Let u = (1, 0, 1, 0)T and v = (1, 1, 1, 1)T . Let S be Span(u, v). Give a basis for the orthogonal complement of S, S ⊥ . 5