advertisement

NAME Id. No, Final Examination, Math 304 Fall 2011, Wilkerson Section. December 13, 2011 – Two Hours – CE 222 No notes, books, calculators, music players, earphones, etc. Show all work. Problem Score I.(30) II.(30) III.(20) IV.(20) V.(15) VI.(20) VII.(15) Total(150) 1. 30 points Let 2 0 0 B = 0 2 1 . 0 1 2 a) (10 pts.) One of eigenvalues of B is 1. Find the other two. 1 b) (10 pts) Find a nonzero eigenvector for each of the three eigenvalues. c) (10 pts) Find an othgononal matrix Q and a diagonal matrix D so that D = QT BQ. 2 2. (30 points) Let 1 1 1 2 1 2 , ~b = . A= 3 1 3 5 1 4 a)(8 points) Write out the normal equations used in solving for the least squares solution to A~x = ~b. b) (12 points) Find the least squares solution of A~x = ~b c) Find an orthonormal basis of the column space R(A) and use this basis to find the projection of ~b onto R(A).(10 points) 3 3.(20 points) Let −4 −3 . C= 10 7 a) Find the eigenvalues of C (10 points). b) (10 points) Find two linearly independent eigenvectors of C. 4 4. (20 points) A linear transformation L : V → V is represented by a matrix after choosing a basis of V . Let VE = R3 with the standard basis E = {e1, e2, e3 }. 0 0 1 Let F be the basis of R3 consisting of v1 = 1 1 , v2 = 1 , v3 = 0 . Let VF denote R3 with 1 1 0 1 0 −1 the F -basis. Suppose that the matrix AE = 2 1 3 represents L in the E-basis. Write out 1 1 0 the linear transformation L as the matrix AF in the F -basis. That is, find AF 5 5. (15 points) Let 1 −1 . A= 1 1 a) (5 points) Find the eigenvalues of A. b) (10 points) Find the singular values of A. 6 6. (20 points) Let 1 2 3 4 A = 2 1 7 4 . 5 7 16 16 a) Row reduce A. (6 points ) b) Give a basis for N(A). (5 points) 1 c) Solve A~x = ~b, for b = 5 for all possible ~x. (9 points) 8 7 1 2 3 7. (15 points) Let S ⊂ R be spanned by the column vectors of B = 5 4 . Find a basis for 9 3 the orthogonol complement of S. 8