麻省理工学院MIT线性代数期末试题

18.06 Professor Strang Final Exam May 20, 2008 Grading 1 2 3 4 5 6 7 8 9 10 Your PRINTED name is: Please circle your recitation: 1) M 2 2-131 A. Ritter 2-085 2-1192 afr 2) M 2 4-149 A. Tievsky 2-492 3-4093 tievsky 3) M 3 2-131 A. Ritter 2-085 2-1192 afr 4) M 3 2-132 A. Tievsky 2-492 3-4093 tievsky 5) T 11 2-132 J. Yin 2-333 3-7826 jbyin 6) T 11 8-205 A. Pires 2-251 3-7566 arita 7) T 12 2-132 J. Yin 2-333 3-7826 jbyin 8) T 12 8-205 A. Pires 2-251 3-7566 arita 9) T 12 26-142 P. Buchak 2-093 3-1198 pmb 10) T 1 2-132 B. Lehmann 2-089 3-1195 lehmann 11) T 1 26-142 P. Buchak 2-093 3-1198 pmb 12) T 1 26-168 P. McNamara 2-314 4-1459 petermc 13) T 2 2-132 B. Lehmann 2-089 2-1195 lehmann 14) T 2 26-168 P. McNamara 2-314 4-1459 petermc Thank you for taking 18.06. If you liked it, you might enjoy 18.085 this fall. Have a great summer. GS 1 (10 pts.) The matrix A and the vector b are A =  1 1 0 2 0 0 1 4 0 0 0 0  b =  3 1 0  (a) The complete solution to Ax = b is x = . (b) ATy = c can be solved for which column vectors c = (c1, c2, c3, c4) ? (Asking for conditions on the c’s, not just c in C(AT).) (c) How do those vectors c relate to the special solutions you found in part (a) ? 2 2 (8 pts.) (a) Suppose q1 = (1, 1, 1, 1)/2 is the first column of Q. How could you find three more columns q2, q3, q4 of Q to make an orthonormal basis ? (Not necessary to compute them.) (b) Suppose that column vector q1 is an eigenvector of A: Aq1 = 3q1. (The other columns of Q might not be eigenvectors of A.) Define T = Q−1AQ so that AQ = QT . Compare the first columns of AQ and QT to discover what numbers are in the first column of T ? 3 3 (12 pts.) Two eigenvalues of this matrix A are λ1 = 1 and λ2 = 2. The first two pivots are d1 = d2 = 1. A =  1 0 1 0 1 1 1 1 0  . (a) Find the other eigenvalue λ3 and the other pivot d3. (b) What is the smallest entry a33 in the southeast corner that would make A positive semidefinite ? What is the smallest c so that A + cI is positive semidefinite ? (c) Starting with one of these vectors u0 = (3, 0, 0) or (0, 3, 0) or (0, 0, 3), and solving uk+1 = 12Auk, describe the limit behavior of uk as k →∞ (with numbers). 4 4 (10 pts.) Suppose Ax = b has a solution (maybe many solutions). I want to prove two facts: A. There is a solution xrow in the row space C(AT). B. There is only one solution in the row space. (a) Suppose Ax = b. I can split that x into xrow + xnull with xnull in the nullspace. How do I know that Axrow = b ? (Easy question) (b) Suppose x∗row is in the row space and Ax∗row = b. I want to prove that x∗row is the same as xrow. Their difference d = x∗row−xrow is in which subspaces ? How to prove d = 0 ? (c) Compute the solution xrow in the row space of this matrix A, by solving for c and d:  1 2 3 1 1 −1 xrow =  14 9  with xrow = c  1 2 3 + d  1 1 −1  . 5 5 (10 pts.) The numbers Dn satisfy Dn+1 = 2Dn − 2Dn−1. This produces a first-order system for un = (Dn+1, Dn) with this 2 by 2 matrix A: Dn+1 Dn  =  2 −2 1 0  Dn Dn−1  or un = Aun−1 . (a) Find the eigenvalues λ1, λ2 of A. Find the eigenvectors x1, x2 with second entry equal to 1 so that x1 = (z1, 1) and x2 = (z2, 1). (b) What is the inner product of those eigenvectors ? (2 points) (c) If u0 = c1x1 + c2x2, give a formula for un. For the specific u0 = (2, 2) find c1 and c2 and a formula for Dn. 6 6 (12 pts.) (a) Suppose q1, q2, a3 are linearly independent, and q1 and q2 are already orthonormal. Give a formula for a third orthonormal vector q3 as a linear combination of q1, q2, a3. (b) Find the vector q3 in part (a) when q1 = 1 2  1 1 1 1  q2 = 1 2  1 −1 1 −1  a3 =  1 2 3 4  (c) Find the projection matrix P onto the subspace spanned by the first two vectors q1 and q2. You can give a formula for P using q1 and q2 or give a numerical answer. 7 7 (12 pts.) (a) Find the determinant of this N matrix. N =  1 0 0 4 2 1 0 3 3 0 1 2 4 0 0 1  (b) Using the cofactor formula for N−1, tell me one entry that is zero or tell me that all entries of N−1 are nonzero. (c) What is the rank of N − I ? Find all four eigenvalues of N . 8 8 (8 pts.) Every invertible matrix A is the product A = QH of an orthogonal matrix Q and a symmetric positive definite matrix H. I will start the proof: A has a singular value decomposition A = UΣV T. Then A = (UV T)(V ΣV T). (a) Show that UV T is an orthogonal matrix Q (what is the test for an orthogonal matrix ?). (b) Show that V ΣV T is a symmetric positive definite matrix. What are its eigenvalues and eigenvectors ? Why did I need to assume that A is invertible ? 9 9 (7 pts.) (a) Find the inverse L−1 of this real triangular matrix L: L =  1 0 0 a 1 0 0 a 1  You can use formulas or Gauss-Jordan elimination or any other method. (b) Suppose D is the real diagonal matrix D = diag(d, d2, d3). What are the conditions on a and d so that the matrix A = LDLT is (three separate questions, one point each) (i) invertible ? (ii) symmetric ? (iii) positive definite ? 10 10 (11 pts.) This problem uses least squares to find the plane C + Dx + Ey = b that best fits these 4 points: x= 0 y = 0 b= 2 x= 1 y = 1 b= 1 x= 1 y = −1 b= 0 x= −2 y = 0 b= 1 (a) Write down 4 equations Ax = b with unknown x = (C,D,E) that would hold if the plane went through the 4 points. Then write down the equations to solve for the best (least squares) solution x̂ = (Ĉ, D̂, Ê). (b) Find the best x̂ and the error vector e (is the vector e in R3 or R4 ?). (c) If you change this b = (2, 1, 0, 1) to the vector p = Ax̂, what will be the best plane to fit these four new points (p1, p2, p3, p4) ? What will be the new error vector ? 11