Math 415 - Midterm 3Thursday, November 20, 2014Circle your section:Philipp Hieronymi 2pm 3pmArmin Straub 9am 11amName:NetID:UIN:Problem 0. [1 point] Write down the number of your discussion section (for instance, AD2or ADH) and the first name of your TA (Allen, Anton, Mahmood, Michael, Nathan, Pouyan,Tigran, Travis).Section: TA:To be completed by the grader:0 1 2 3 4 5 6 ShortsP/1 /? /? /? /? /? /? /? /?Good luck!1Instructions• No notes, personal aids or calculators are permitted.• This exam consists of ? pages. Take a moment to make sure you have all pages.• You have 75 minutes.• Answer all questions in the space provided. If you require more space to write youranswer, you may continue on the back of the page (make it clear if you do).• Explain your work! Little or no points will be given for a correct answer with noexplanation of how you got it.• In particular, you have to write down all row operations for full credit.Problem 1. Let A =1 −11 01 11 2and b =50510. Find a least squares solution of Ax = b.Problem 2. Let W = span0101,0111.(a) Find an orthonormal basis for W .(b) What is the orthogonal projection of1210onto W ?(c) Write1000as the sum of a vector in W and a vector in W⊥.(d) Find the projection matrix corresponding to orthogonal projection onto W .Problem 3. Find the QR decomposition of A =4 25 00 0 −23 −25 0.Problem 4. Find the least squares line for the data points (1, 1), (2, 1), (3, 4), (4, 4).Problem 5.(a) Let A =a b cd e fg h i. Write down the cofactor expansion of det(A) along the secondcolumn.(b) Let A = [a1a2a3] and B = [b1b2b3] be two 3 × 3-matrices. Suppose that det(A) = 5and b1= a1, b2= a1+ 2a2, b3= a3. What is det(B)?(c) Find det1 1 1 11 1 4 41 −1 2 −21 −1 8 −8.2(d) Find det0 0 3 10 0 2 22 −1 1 11 0 −1 0.Problem 6. Let A =−1 0 1−3 4 10 0 2.(a) Find the eigenvalues of A, as well as a basis for the corresponding eigenspaces.(b) Diagonalize A. (That is, write A = P DP−1where D is diagonal.)Problem 7. Consider the vector spaceV = {f : R → R : f is 7-periodic and f is “nice”}.Here, f “nice” means, for instance, that f should be piecewise continuous or (more generally)thatR70f(t)2dt should be finite.(a) What is a natural inner product on V ?(b) Consider the 7-periodic function f(t) with f(t) =1, for 0 ≤ t < 3,2, for 3 ≤ t < 7.Compute the orthogonal projection of f (t) onto the span of cos32πt7.Problem 8. Consider the space P2of polynomials of degree up to 2, together with the innerproducthp(t), q(t)i =Z10p(t)q(t)dt.(a) Is the standard basis 1, t, t2an orthogonal basis?(b) Apply Gram–Schmidt to 1, t, t2to obtain an orthonormal basis of P2.(c) What is the orthogonal projection of t2onto span{1, t}?3SHORT ANSWERSNote: On the actual exam all short answer question will be multiple choice. You will be en-tering your answers to the multiple choice questions on a scantron sheet that will be includedwith your exam. So please bring a Number 2 pencil to the exam. Thanks.Short Problem 1. If A and B are 3 × 3 matrices with det(A) = 4 and det(B) = −1. Whatis the determinant of C = 2ATA−1BA?Short Problem 2. If A is an n × n matrix, and S is an invertible n × n matrix. Are thecharacteristic polynomial of A and SAS−1equal? The determinant?Short Problem 3. Let A be a 7 × 7 matrix with dim Nul(A) = 1. What can you say aboutdet(A)?Short Problem 4. Consider A =1 1 0 01 −1 0 00 0 1 10 0 1 −1.(a) Using that the columns of A are orthogonal, find A−1.(b) Let w1, . . . , w4be the columns of A. Without solving equations, find coefficientsc1, . . . , c4such that1234= c1w1+ . . . + c4w4.Short Problem 5. Let A be a n ×n matrix with AT= A−1. What can you say about det(A)?Short Problem 6. Let A be an n × n matrix with eigenvalue λ. Determine whether each ofthe following statements is correct.(a) λ2is an eigenvalue of A2.(b) λ−1is an eigenvalue of A−1.(c) λ + 1 is an eigenvalue of A + I.(d) λ cannot be zero.Short Problem 7. Consider a matrixQ =1√31√14?1√32√14?1√3−3√14?,in which the third column has not been specified, yet. Which of the following vectors can beadded as a third column of Q such that Q is orthogonal?(a)−541,(b)−5√424√421√42,(c)001,4(d) none of the above.Short Problem 8. True or false?(a) If ATA is diagonal, then A has orthogonal columns.(b) If A is an orthogonal matrix, then ATis an orthogonal matrix.(c) If Ax = 0, then x is orthogonal to the columns of A.(d) For all n × n matrices A and B, det(AB) = det(A) det(B).(e) For all n × n matrices A and B, det(A + B) = det(A) + det(B).(f) Every orthonormal set of vectors is linearly independent.(g) Every subspace of Rnhas an orthogonal basis.(h) If every row of A adds up to 0, then det(A) = 0.(i) If every row of A adds up to 1, then det(A) = 1.(j) If A is invertible and B is not invertible, then AB is invertible.(k) The determinant of A is the product of the diagonal entries of A.Short Problem 9. Suppose that the projection matrix corresponding to orthogonal projectiononto V is P =13029 −2 5−2 26 105 10 5.(a) Is v =−210in V ?(b) Find the vector in V which is closest to w =110.(c) What is the dimension of V
View Full Document
Unlocking...