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Winter2003 - Faculty of Mathematics QXV fl University of...

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Unformatted text preview: Faculty of Mathematics QXV fl University of Waterloo Maw/t V“ MATH 136 — Winter 2003 FINAL EXAMINATION Date: April 167 2003 NO AIDS, NO CALCULATORS Time: 9:00 — 12:00 Instructors: (Please check your section) Section 1 K. Zhao 2 C.Y. Lam 3 K. Zhao 4 F. Szechtman 5 K.J. Giuliani 6 CG. Hewitt DDDDDDD 7 H. Muzaffar Family Name: (please print) _____________________.___ Initials: Id. Number: Signature: _____________________ This exam has 14 pages, please check to make sure that you have them all. Your mark will be affected by how clearly you present your solutions. ‘ Math 136 — Final Exam Winter 2003 1. a) Find conditions on the parameter p E R for Which the system of equations 1.m+2.y+3.z = ——1 2.x+5.y+(p2~—13)z=p+7 2.$+4.y+(p2— 11)z=p+3 has (i) No solutions (iii) A unique solution (iii) More than one solution. (i) No solution for: (ii) A unique solution for: (iii) More than one solution for: b) Find the rank and nullity of A and AT when “>me mAww CbU‘rhC/O «lama; 00-40501 Page 2 of 14 Math 136 - Final Exam Winter 2003 2. Let S = {V1V2, . . .vp} be a subset of a vector space V. a) Define what it means for S to be linearly independent. b) Define Span(S). c) Define what it means for S to be a basis for V. (1) Let v 6 V and suppose that S is a basis for V. Prove that v may be written as a unique linear combination of the elements of S. Page 3 of 14 Math 136 - Final Exam . Winter 2003 1 3. Let A be the matrix A = 0 1 a) Evaluate A‘l. b) Express A as the product of elementary matrices. Page 4 of 14 Math 136 — Final Exam Winter 2003 4. In P2(R) consider the two bases 3 = {t2 + 1,1? — 2,t + 3} and T = '{2t2 + t,t2 + 3, t}. a) Find Ps._T, the transition matrix from T to S. You may use Q3a if you Wish. b) If v = 8t2 —— 4t find MT. 0) Use your solution to a) to evaluate [v]5. Page 5 of 14 Math 136 — Final Exam Winter 2003 5. Let A,B e ManOF). a) Prove that if A and B are invertible then AB is invertible. Let C E MnxnflF). We say that C is nilpotent if 0’“ = On for some positive integer k. Note that 0’“ = 0.0. . . C (k times). 0 1 1 b) Show that C: 0 0 0 1 is nilpotent. 0 0 c) Prove that every nilpotent matrix is non-invertible. Page 6 of 14 Math 136 — Final Exam Winter 2003 6. Let V = nggflR) and B E V is a fixed matrix. Let 53 = {A E V : AB = 2BA}. a) Prove that $3 is a vector subspace of V. b) Give a B such that 33 = V. c) Give a B such that $3 = {(3 3) }. Page 7 of 14 Math 136 - Final Exam Winter 2003 7. DBIBIIIHHB Whether or not the given function is an inner product on the given vector space. a) V = R2,<(x1>, <91>>= x? + mg + y? + y§ where x1,x2,y1,y2, E 1R- 132 112 z w _ _ b) V = C2, <( 1) , (“13> = zlwl + 222112 — zlwz — 221111 where 21, 22,w1,w2 E C. (I; _ c) V = 1R2, <<m:) , (39> = 2:131:11 +4z2y2+2m1y2+2x2y1 where 11:1, $2,311, 112 6 R Page 8 of 14 Math 136 - Final Exam Winter 2003 8. a) Let (V, <, >) be an inner product space and W be a vector subspace of V. Define WJ'. b) Find an orthonormal basis for the subspace, S, of R3 given by S = {(23, y, z) E R3 : a: — 2g + 3z = 0}. Use the standard inner product. c) Find an orthonormal basis for 5*. Page 9 of 14 Math 136 — Final Exam Winter 2003 9. Consider the vector space of continuous real valued functions defined on the interval [—1, 1], V = C'[—1, 1]. Let the inner product on V be given by <f<t>,g<t>> = /—1f(t)g(t)dt' Given that B = { «i5, cos 7rt, sin m} is an orthonormal basis for a subspace S of V, use the following information to find: (i) the projection of the function f (t) = 132 onto S, and (ii) find the length of this projection. 1 1 1 Given Information: / tzdt = 2/3, / t2 cos 7rt dt = —4/7r2, / t2 sin 7rt dt = 0. —1 —1 —1 Page 10 of 14 Math 136 - Final Exam Winter 2003 10. Consider the following 5 data points: (—2,0), (~1,1), (0,1), (1,0), (2,1). Use the method of least squares to determine the equation of the line which is the best fit to the given data. That is, find a, b, E R such that y = a3: + b is the best fit to this data. Page 11 of 14 Math 136 — Final Exain Winter 2003 11. let S = {V1,V2, . . . VP} be an orthogonal set of non-zero vectors in an inner product space (V, <, >). Prove that S is linearly independent. Page 12 of 14 Math 136 - Final Exam Winter 2003 12. Let V be a finite-dimensional vector space and (V, <, >) be a real inner product space. Suppose {V1, v2, . . . ,vp} is an orthonormal set of vectors in V and x E V. :2 Prove that H x ”22 Z(x,vi)2. i=1 Page 13 of 14 ' Math 136 — Final Exam Winter 2003 ROUGH WORK Page 14 of 14 ...
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