Practice final from math 311, Cheat Sheet of Mathematics

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MATH 311 — Practice Final II

MATH 311 — Practice Final Il MATH 311, Section 5000 Final Exam May 32, 3026 Name:, FORM A Student Number: This exam has 15 questions for a total of 110 points. Show all your work! In order to obtain full credit for partial credit problems, all work must be shown. Credit will not be given for an answer not supported by work. For other problems, points might be deducted, at the sole discretion of the instructor, for an answer not supported by a reasonable amount of work. The point value for each question is in parentheses to the right of the question number. YOU MAY NOT USE A CALCULATOR ON THIS EXAM. PLEASE PUT AWAY YOUR CELL PHONE. Do not write in this box. 11, 12: 13: 14: Total: MATH 311 FINAL EXAM Form A May 32, 3026 1. (5 points) Determine if the following statement is TRUE or FALSE: If a system Ax = b has a unique solution for some non-zero b, then so does the system Aw = 0. (a) TRUE (b) FALSE 2. (5 points) Determine if the following statement is TRUE or FALSE: If {v,u,w} are an or- thonormal set of vectors in R", then there span is a three dimensional subspace of R”. (a) TRUE (b) FALSE 3. (5 points) Determine if the following statement is TRUE or FALSE: Every vector field X on R3 is the curl of another vector field Y, ie. we can always find another Y satisfying X = V x Y. (a) TRUE (b) FALSE 4. (5 points) Determine if the following statement is TRUE or FALSE: If A is m x n and B is px mand nullity(A) = k then nullity(BA) > k. (a) TRUE (b) FALSE Page 3 of 10 MATH 311 FINAL EXAM Form A May 32, 3026 7. (5 points) Let S be a bounded region of the plane with area 7 and R,,/3 be the counterclock- wise rotation by angle 7/3, T be a transformation whose matrix A satisfies that AB = BA where B has distinct eigenvalues 2 and 3 and A has only the eigenvalue 3, and R,/¢ be the counterclockwise rotation by angle 7/6. Then the area of the region R,/g6T'Rz/3(9) is 2 BY a) b) (c) (¢) 3 Pe el wn ¥ sl 8. (5 points) The vector [10 12)" written as the sum of two vectors, one on the line {(x,y) : y= 2x} and one on the line {(x,y) : y = v/2}, is (a) [10 12)7= 41 2]7+8[2 1)” (b) [10 12]7= = fa 2]7+8[2 1] (c) [10 12)"= 4f1 2)7-8[2 1)”, (a) [10 12)" = 81 2)" 444 [2 a)”. Page 4 of 10 MATH 311 FINAL EXAM Form A May 32, 3026 9. (5 points) The solution set to the equation 123 -2 0 3 4)%> |-4 24 6 —4 is given by ) ) A line through the vector [1 0 — 1]? (c) A two dimensional plane through the vector {1 0 — 1]” ) 10. (5 points) Suppose @ is an eigenvector of an n x n matrix A corresponding to an eigenvalue and P is an invertible n x n matrix. Then Pz is also an eigenvector of P(eI — A)(A8I — \?A)(sin(A)I — A)P~!. What is the corresponding eigenvalue of Pa for P(eI — A)(8I — d7A)(sin(A)I — A)P7!? (a) e*A3 sin(A) (b) e + A3 + sin(A) (c) e +8 — A? + sin(A) (a) ) (c Q ot enough information to determine. Ze Page 6 of 10 MATH 311 FINAL EXAM Form A May 32, 3026 12. (10 points) Compute the line integral of the vector field 3yx? + 23 along the triangle in R® with vertices (0,0,3), (0, 2,0), (4,0,0) oriented so that if one starts at (0,0, 3) the next vertex is (4,0,0). Page 7 of 10 MATH 311 FINAL EXAM Form A May 32, 3026 13. (10 points) Consider the following 3 x 3 matrix A given by 1 1 -v2 -v2 0 1 (a) (5 points) Find all eigenvalues of A. Is A orthogonally diagonalizable? Why or why not? HINT: you got this *sunglasses*. (b) (5 points) What 3 x 3 matrix does the following matrix A’ — 2A° — 2A? + A equal? Page 9 of 10 MATH 311 FINAL EXAM Form A May 32, 3026 15. (10 points) Suppose that A is a diagonalizable n x n matrix, and for each eigenvalue A; of A let Ey, denote the \j-eigenspace of A (here 1 < i < p R” denote the linear transformation Pj(a) = #;. Show that P? = P, and P;o Pj =0ifi Fj. (c) (3 points) Show that if T denotes the linear transformation T(a) = Aw, then T = \,P; + sts ApPp.