Math 51 - Winter 2009 - Final ExamName:Student ID:Select your section:Penka Georgieva Anssi Lahtinen Man Chun Li Simon Rubinstein-Salzedo02 (11:00-11:50 AM) 03 (11:00-11:50 AM) 12 (1:15-2:05 PM) 17 (1:15-2:05 PM)06 (1:15-2:05 PM) 11 (1:15-2:05 PM) 08 (11:00-11:50 AM) 21 (11:00-11:50 AM)Aaron Smith Nikola Penev Eric Malm Yu-jong Tzeng09 (11:00-11:50 AM) 14 (1:15-2:05 PM) 15 (11:00-11:50 AM) 51A20 (10:00-10:50 AM) 24 (2:15-3:05 PM) 23 (1:15-2:05 PM)Signature:Instructions: Print your name and student ID number, print yoursection number and TA’s name, write your signature to indicate thatyou accept the honor code. During the test, you may not use notes,books, calculators. Read each question carefully, and show all yourwork.There are ten problems on the pages numbered from 1 to 13, with thetotal of 140 points. Point values are given in parentheses. You have 3hours (until 10PM) to answer all the questions.In the exam all vectors are columns, but sometimes we use transpose to write them horizon-tally.Thus v =v1v2...vk= [v1, v2, . . . , vk]T.Similarly vTis a row [v1, v2, . . . , vk].The dot product of two vectors is denoted as v · w.Problem 1. For what values of parameters a and b does the systemx + 2y + 3z = 12x + 4y + a · z = ba) (3 pts.) Has more than one solution?b) (3 pts.) Has unique solution?c) (3 pts.) Has no solution?1Problem 2. (6 pts.) Find det A ifA−1B =1 12 340 6 130 0 23and det B = 23.Problem 3. a) (6 pts.) Write an equation of the plane in R3that is passing through thepoints (−1, 1, 0), (2, −3, 1) and (2, 3, −2).2b) (6 pts.) Let T be the linear transformation given by multiplication byA =3 4 170 −3 230 0 −1and let R be a triangle in R3whose area is 3. Find the area of the region T(R).Problem 4. Let B = {v1, v2, v3} be a basis of a linear subspace V of Rn.a) (6 pts.) Show that if x · vi= 0 for each i = 1, 2, 3, then x · v = 0 for any v ∈ V .3b) (6 pts.) Let V⊥be the set of vectors orthogonal to all the vectors of V , i.e.:V⊥= {x ∈ Rn; x · v = 0 for all v ∈ V }.Find a matrix A such that N (A) = V⊥. (Your answer should use the vectors v1, v2, v3.)c) (6 pts.) Show that the only vector belonging simultaneously to V and V⊥is the zerovector. (Hint: consider v · v for such a vector v).4Problem 5. Let v =111∈ R3and L = span(v). Let P be the orthogonal projection onthe line L in R3.a) (8 pts.) If S = {x ∈ R3; x · v = 0}, show that S is a subspace of R3. Check all threeconditions of the linear subspace.5b) (8 pts.) Find a basis {v1, v2} of S.c) (4 pts.) Given that B = {v, v1, v2} is a basis of R3(v1, v2are the vectors you foundabove), find the matrix of P in B?6Problem 6. a) (8 pts.) Find the points on the sphere x2+ y2+ z2= 24 where f(x, y, z) =2x + y − z has its minimum and maximum values.b) (3 pts.) What is the geometric meaning of the minimum and maximum points?7Problem 7. a) (9 pts.) Find and classify the critical points of the functionf(x, y) = 3y2− 2y3− 3x2+ 6xyb) (4 pts.) Is it possible that f (x, y) has a global minimum or maximum?8c) (5 pts.) Find an equation of the tangent plane to the graph of f(x, y) at the point(2, 2, 8). (you do not need to verify that f(2, 2) = 8.)Problem 8. Let A be the matrixA =7 5 −7−5 −3 61 1 0a)(5 pts.) Find eigenvectors corresponding to the eigenvalue λ = 1.9b) (5 pts.) Given that v = [−1, 1, 0] is an eigenvector of A =7 5 −7−5 −3 61 1 0, find thecorresponding eigenvalue.c) (5 pts.) Given that the characteristic polynomial of the matrix A is(λ − 2)(λ − 1)2determine if the matrix A is diagonalizable. Justify your answer.10d) (4 pts.) Using the same characteristic polynomial (λ−2)(λ−1)2, determine if the matrixA is invertible. Justify your answer.Problem 9. Let f be the following function on R3:f(x, y, z) = (x + y2, y + z2, z + x2)and let g(x, y, z) = exp(x + y + z).a) (5 pts.) Show that the matrix of the derivative of f is:D =1 2y 00 1 2z2x 0 1.11b) (5 pts.) Starting from the point (0, 0, 0) ∈ R3in which direction shall one move inorder to increase g(x, y, z) fastest?c) (7 pts.) Calculate the derivative of g ◦ f.12Problem 10. Let A be the following matrix:A =1 2 y2 x 11 2 2y.a) (4 pts.) Calculate det(A).b) (6 pts.) Is the determinant more sensitive to changes in x or y near x = 1 and y = 0?13Problem Score Maximum1 92 63 124 185 206 117 188abc 158d 49a 59bc 1210 10Total
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