Math 51 - Winter 2009 - Midterm Exam IIName: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,bo oks, calculators. Read each question carefully, and show all yourwork.There are nine problems on the pages numbered from 1 to 9, with thetotal of 100 points. Point values are given in parentheses. You have 2hours (until 9PM) 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. (10 pts.)a) Let [1, 2, 3] be the first row of a square matrix A anda−11be the first column of asquare matrix B. Find a if you know that B is the inverse of A.b) Let [1, 2, 3] be the first row of a square matrix A anda−11be the second column of asquare matrix B. Find a if you know that B is the inverse of A.Problem 2. (10 pts.) Find the inverse of the matrix1 2 30 1 40 0 11Problem 3. (10 pts.) Assume that A is an invertible matrix.a) If Ax = u, what is A−1(2u)?b) I f Ax1= 2u1and Ax2= 3u2, what is A−1(3u1− 8u2)?2Problem 4. Let T : R3→ R3be the following linear map:Tx1x2x3=2x2− x33x1− 2x2−2x1+ 2x2+ x3.a) (4 pts.) Write the matrix A of T (in the standard coordinates).b) (6 pts.) Given that the characteristic polynomial of the above matrix A isp(t) = (1 − t)(t − 2)(t + 4)(you do not need to verify that), find a basis of eigenvectors of T.3c) (4 pts.) Is T diagonizable? Explain your answer.Problem 5. (10 pts.) Find a matrix C such that1 00 4= C2 00 2C−1or explain why such matrix does not exist.4Problem 6. Let A be the following 3 × 3 matrix:A =2 1 −1−1 2 11 2 0with its inverseA−1=−2 −2 31 1 −1−4 −3 5(you do not need to verify that).a) (4 pts.) Solve the systemA · x =1−13b) (4 pts.) Find a matrix M that has eigenvectors v1=2−11and v2=122, b othcorresponding to the eigenvalue 0, and an eigenvector v3=−110corresponding to theeigenvalue 1.5c) (4 pts.) Find the inverse M−1of the matrix M described in part b) or explain why doesthe inverse fail to exist.d) (4 pts.) Let T : R3→ R3be a linear map whose matrix in standard coordinates is thematrix M described in part a). Describe the map T in terms of “projections”, “reflections”and/or “rotations”. Be specific as to which planes, lines or points you are using in theprojection, reflection or rotation.6Problem 7. (10 pts.) Find an equation of the tangent plane to the graph of the functionf(x, y) = 2x3y2− 3y2x + 3at the point (1, 1, 2).Problem 8. (10 pts.) Let T : R2→ R2be a linear map such thatT (e1+ e2) =12and T (e1− e2) =34a) Find T (e1).b) Find the matrix (in standard basis) of the map T .7Problem 9. (10 pts.) Let Q(x, y) = x2− 2xy + ay2a) For what values of the parameter a is the quadratic form Q positive definite?Hint, which you may ignore: write Q(x, y) as (x − y)2+ . . . · y2and use the definition ofbeing positive definite.b) For what values of the constant a does the function Q(x, y) satisfy the following expression:∂2Q∂x∂y=∂2Q∂y28Page Score Maximum1 202 103 104 145 86 87 208 10Total
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