Name: Student ID: Section: 2Prof. Alan J. Laub June 9, 2008Math 33A – FINAL EXAMINATIONSpring 2008Instructions:(a) The exam is closed-book (except for one page of notes) and will last two and a half(2.5) hours.(b) Notation will conform as closely as possible to the standard notation used in thelectures.(c) Do all 6 problems. Problems 1–5 are worth 15 points each; problem 6 is worth 25points (5 subparts, each worth 5 points). Some partial credit may be assigned ifwarranted.(d) Label clearly the problem number and the material you wish to be graded.1. Consider the initial-value problem˙x(t) = Ax(t); x(0) = x0for t ≥ 0. Let A =·−7 5−10 8¸and x0=·10¸. Determine an explicit expression forx(t), i.e., solve the initial value problem.2. Let A ∈ IRm×nand define the three matrices Bk= I − kAA+for k = 1, 2, 3.(a) Which of B1, B2, B3are symmetric?(b) Which of B1, B2, B3are idempotent (i.e., B2k= Bk)?(c) Which of B1, B2, B3are orthogonal?3. (a) For what values of the scalar α, if any, is A =1 2 32 α 43 4 5positive definite?Show your reasoning carefully.(b) For what values of the scalar β, if any, is B =β −1 −1−1 β −1−1 −1 βpositive definite?Show your reasoning carefully.4. Let A =4 0 −20 1 01 0 1. Find a nonsingular matrix X that makes A similar to adiagonal matrix (and find the diagonal matrix) or, alternatively, show that such amatrix X does not exist. Verify the formulas Tr(A) = λ1+ λ2+ λ3and det(A) =λ1λ2λ3.5. Let A =1 12 22 2and suppose you have found (e.g., using Matlab) an SVD of Aof the form13−2√552√51523√554√515230 −√533√2 00 00 0"√22√22√22−√22#=1323233√2h√22√22i.Many things about A as a linear transformation are available immediately (or via abrief calculation) from its SVD. In the following, you are asked to determine a few ofthem. Any attempt at any other method of determining them will receive zero credit.(a) Determine orthonormal bases for each of the four fundamental subspaces of A.(b) Determine the rank(A).(c) Determine the orthogonal projections PKer(A)and PIm(A).(d) Determine A+.6. Determine the best answer in each case:(a) Suppose V ∈ IRn×khas k orthonormal columns where 0 < k < n. Then thevalue of det·InVVT0¸is(A) 1 (B) −1 (C) 0 (D) (−1)k(E) none of these(b) The eigenvalues of A =·0 4−4 0¸are(A) 2i and −2i (B) 4i and −4i (C) 2 and −2 (D) 4 and −4 (E) none of these(c) Suppose A ∈ IRn×nis symmetric and negative definite. Suppose further that Vis an orthogonal matrix that diagonalizes A, i.e., VTAV = Λ = diag(λ1, ··· ,λn)where λiare the eigenvalues of A. Then an SVD of A is given by(A) V ΛVT(B) (−V )ΛVT(C) V (−Λ)VT(D) V Λ(−V )T(E) none of these(d) What is the geometric multiplicity of the eigenvalue 0 of the matrixA =2 2 ··· 22 2 ···...............2 ··· ··· 2∈ IRn×n?(A) 1 (B) 0 (C) n (D) n − 1 (E) none of these(e) Suppose A ∈ IRm×nhas an SVD given by UΣVT= [U1U2]·S 00 0¸·VT1VT2¸.Then the general solution of Ax = b (with b ∈ Im(A)) is given by(A) x = V1S−1UT1b + V2z where z ∈ IRn−ris arbitrary(B) x = U1SVT1b + V2z where z ∈ IRn−ris arbitrary(C) x = V1S−1UT1b(D) x = U1SVT1b(E) none of