Name: Student ID: Section: 2Prof. Alan J. Laub June 11, 2009Math 33A – FINAL EXAMINATIONSpring 2009Instructions:(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. In a certain experiment, the following measurements (ti, yi) are taken:(−2, 7), (−1, 2), (0, 1), (1, 6), (2, 15) .It is desired to “fit” this data set with a function of the formy = αt + βt2where α and β are parameters to be determined. Compute the best parameters α andβ (in the 2-norm sense) using the method of linear least squares.12. Consider the initial-value problem˙x(t) =dx(t)dt= Ax(t); x(0) = x0for t ≥ 0. Let A =·0 1−12 −7¸and x0=·11¸.(a) Is A asymptotically stable? Why or why not?(b) Determine an explicit expression for x(t), i.e., solve the initial value problem.23. For what values of the scalars a and b, if any, is the matrix A =a b 0b a b0 b apositivedefinite? Show your reasoning carefully, i.e., state explicitly which criteria or criterionfor positive definiteness you are using.4. In class we saw that if A ∈ IRm×nr, m ≥ n, was factored as A = QR where Q hadorthonormal columns and R ∈ IRn×nnwas upper triangular (for example, via Gram-Schmidt orthogonalization), then the solution of the linear least squares problem wasgiven byx = (ATA)−1ATb = (RTQTQR)−1RTQTb = R−1QTb.In other words, A+= R−1QT. Show that this must be so by verifying the four Penroseconditions.35. Let A =1 2 −20 5 −40 6 −5.(a) Find a nonsingular matrix S that makes A similar to a diagonal matrix (and findthe diagonal matrix) or, alternatively, show that such a matrix S does not exist.(b) Verify the formulas Tr(A) = λ1+ λ2+ λ3and det(A) = λ1λ2λ3.46. Let A =1101 2 11 2 21 2 31 2 4and suppose you compute (e.g., using Matlab) an SVDUΣVTof A to produce the following matrices:U =−0.3147 −0.7752 0.5000 0.2236−0.4275 −0.3424 −0.8333 0.0745−0.5402 0.0903 0.1667 −0.8199−0.6530 0.5231 0.1667 0.5217, Σ =0.6922 0 00 0.1445 00 0 00 0 0,V =−0.2796 −0.3490 0.8944−0.5592 −0.6981 −0.4472−0.7805 0.6252 0.0000.Suppose, furthermore, that you also compute, from this data, thatA+=2 1 0 −14 2 0 −2−3 −1 1 3.As you know, many things about A as a linear transformation are available immedi-ately (or via a brief calculation) from its SVD. In some of the following, you are askedto determine a few of them. Any attempt at any other method of determining themwill receive zero credit.(a) Determine the rank of A.(b) Draw a diagram of the four fundamental subspaces of A and indicate the dimen-sions of each subspace.(c) Determine orthonormal bases for each of the four fundamental subspaces of A.(d) Determine the orthogonal projections PR(A)and PN (A).(e) Find the minimum (2-)norm solution of the linear least squares problemminxkAx − bk2where b =