Name: Student ID: Section: 2Prof. Alan J. Laub May 16, 2008Math 33A – MIDTERM EXAMINATION IISpring 2008Instructions:(a) The exam is closed-book (except for one page of notes) and will last 50 minutes.(b) Notation will conform as closely as possible to the standard notation used in thelectures.(c) Do all 5 problems. Each problem is worth 20 points. Some partial credit may beassigned if warranted.(d) Label clearly the problem number and the material you wish to be graded.1. Suppose x, y ∈ IRn. Use the determinant formulas given in class to show thatdet(In− xyT) = 1 − yTxwhere Indenotes the n × n identity matrix.Hint: Recall that the determinant formulas involve computing det·A BC D¸. Evalu-ate this formula two different ways for the same appropriate values of A, B, C, D. Youwill receive no credit for other ways of trying to evaluate the determinant.2. In a certain experiment, the following measurements (ti, yi) are taken:(−2, 4), (−1, 0), (0, −2), (1, −1), (2, 6)It is desired to “fit” this data set with a function of the formy = αt2+ βwhere α and β are parameters to be determined. Compute the best parameters αand β using the method of linear least squares (by solving the normal equations sinceyour calculations will be done by hand!).3. Suppose that for A =2 12 11 5you have found the following QR factorization of A:A = [a1, a2] = QR =23−13√223−13√21343√2·3 30 3√2¸.Show how to get the unit Gram-Schmidt vectors q1and q2of the form q1= αa1,q2= βa1+ γa2with q1⊥ q2from this factorization, i.e., find α, β, γ.4. Let A =1 11 10 0. Find the unique decomposition of y =357into y = v + w withv ∈ Im(A) and w ∈ Im(A)⊥. You will find the formula A+=·1414014140¸useful.5. Suppose A ∈ IRm×nnhas a QR factorization A = Q1R, where Q1∈ IRm×nhas or-thonormal columns and R ∈ IRn×nnis upper triangular. Show, by verifying the fourPenrose conditions, that A+=