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06aut-f - MATH 51 Instructions: FINAL EXAM December 11,...

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MATH 51 FINAL EXAM December 11, 2006 Instructions: No calculators, books, notes, or electronic devices may be used during the exam. You have 3 hours. There are 16 problems, each with multiple parts. Many questions have short answers requiring no computation. The point value of each part of each problem is indicated in brackets at the beginning of that part. You should work quickly so as to not leave out problems towards the end of the exam. Show computations on the exam sheet. If extra space is needed use the back of a page. Name: (print clearly) Signature: (for acceptance of honor code) Your TA/discussion section (circle one): Antebi (15, 18) Ayala (3, 6) Easton (14, 17) Fernanadez (2, 5) Kim (8, 11) Koytcheﬀ (9, 12) Lo (21, 24) Rosales (26, 27) Tzeng (20, 23) Zamﬁr (29, 30) Schultz (51A) Problem 1 (10 points) Problem 2 (10 points) Problem 3 (10 points) Problem 4 (10 points) Problem 5 (10 points) Problem 6 (10 points) Problem 7 (10 points) Problem 8 (10 points) Problem 9 (10 points) Problem 10 (10 points) Problem 11 (10 points) Problem 12 (10 points) Problem 13 (10 points) Problem 14 (10 points) Problem 15 (10 points) Problem 16 (10 points) Total (160 points)

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1. Suppose rref( A ) = 1 2 0 1 2 0 0 1 1 2 0 0 0 0 0 and suppose you know that A 1 2 3 4 - 5 = - 1 7 9 . (a) [5] Write in parametric form all solutions of the system of equations A x = - 1 7 9 . (b) [5] Denote the i -th column of A by a i . Suppose a 2 = 2 4 6 and a 4 = 1 - 1 - 1 . Find A . [Hint: Make use of some linear dependence relations between the columns of A .] 1
2. For each of the following subsets S of R 3 determine if S is a subspace of R 3 . If not , give a reason. If S is a subspace you don’t need to prove that, but give a basis of S . (a) [2] S = x y z ± ± ± ± ± x - 2 y + 3 z = 2 (b) [4] S = All x y z orthogonal to both 1 2 0 and 0 - 1 3 (c) [4] S = span 1 0 - 1 , 0 1 - 1 , 3 2 - 5 2

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3. (a) [4] For which choice(s) of constant k is the matrix 0 1 1 1 2 k 1 4 k 2 not invertible? (b) [3] Let A = ± cos θ - sin θ sin θ cos θ ² . Find det( A ) and A - 1 .
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06aut-f - MATH 51 Instructions: FINAL EXAM December 11,...

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