Linear and Nonlinear Computation Models (CSE 4190.313) Midterm Exam: April 21, 2014 1. (15 points) If A has row 1 + row 2 = row 3, show that A is not invertible: (a) (5 points) Explain why Ax = (1, 0, 0)T cannot have a solution. (b) (5 points) Which right-hand sides (b1 , b2 , b3 ) might allow a solution to Ax = b? (c) (5 points) What happens to row 3 in elimination? Solution: (a) Suppose there exists a solution x such that Ax = (1, 0, 0)T , T a1 T where A = a2 and aT3 = aT1 + aT2 . aT3 Then aT1 · x = 1, aT2 · x = 0, and aT3 · x = 0. But, aT3 · x = (aT1 + aT2 ) · x = aT1 · x + aT2 · x = 1 + 0 = 1 6= 0 = aT3 · x, which is a contradiction. (b) To be consistent with the left-hand side, we have b1 + b2 = b3 . (This is because b1 + b2 = aT1 · x + aT2 · x = aT3 · x = b3 .) a11 a12 a13 0 0 aT1 T ∗ ∗ 1 0 a2 = 0 (c) −l21 0 0 0 aT3 −1 −1 1 1 a11 a12 a13 1 0 0 aT1 T ∗ ∗ 1 0 A = a2 = l21 = LU 0 T 0 0 0 1 + l21 1 1 a3 Thus, the third row is reduced to a zero vector (0, 0, 0). 2. (10 points) If Ax = b1 has infinitely many solutions, why is it impossible for Ax = b2 (new right-hand side) to have only one solution? Could Ax = b2 have no solution? Solution: (a) The columns of A are linearly dependent. Thus, there are infinitely many nonzero xN (6= 0) ∈ N (A) with AxN = 0. Now, for the unique solution x for Ax = b2 , we have Ax = A(x + xN ) = Ax + AxN = b2 + 0 = b2 , for infinitely many x = x + xN . This is a contradiction. (b) Ax = b2 has no solution if and only if b2 6∈ C(A). 3. (10 points) If v1 , · · · , vn is an orthonormal basis for Rn , show that v1 v1T + · · · + vn vnT = I. Solution: (v1T · x)v1 + · · · + (vnT · x)vn v1 (v1T · x) + · · · + vn (vnT · x) (v1 · v1T ) · x + · · · + (vn · vnT ) · x (v1 · v1T + · · · + vn · vnT ) · x, for all x ∈ Rn . Ix = x = = = = 4. (15 points) (a) (5 points) How far is the plane x1 + x2 − x3 − x4 = 8 from the origin, and what point on it is nearest? (b) (5 points) Is there a matrix whose row space contains (1, 1, 0) and whose nullspace contains (0, 1, 1)? (c) (5 points) If A is a square matrix, show that the column space of A2 is contained in the column space of A. (a) The plane can be represented as 21 (x1 +x2 −x3 −x4 ) = 4 with a unit normal n = 12 (1, 1, −1, −1), which is at distance 4 from the origin. This plane contains a point p0 = (8, 0, 0, 0) which can ˆ 0 on the orthogonal line passing through the origin (and thus parallel be projected to a point p ˆ 0 = (p0 · n)n = 4 · 12 (1, 1, −1, −1) = to the normal direction n). The projection point is p (2, 2, −2 − 2), which is the nearest point of the plane to the origin. (b) The row space and the nullspace are orthogonal complements each other. But the two vectors are not orthognonal as their inner product (1, 1, 0) · (0, 1, 1) = 1 6= 0. Thus, there is no such a matrix. (c) A2 = AA = A [a1 · · · an ] = [Aa1 · · · Aan ]. Each column Aaj of A2 can be represented as a P linear combination of the columns of A: Aaj = ni=1 aij ai , where aj = (a1j , · · · , an j)T . Thus, we have C(A2 ) ⊂ C(A). 5. (15 points) Without computing A, find bases for the four fundamental subspaces: 1 0 0 1 2 3 4 A = 6 1 0 0 1 2 3 . 9 8 1 0 0 1 2 (a) Basis of C(A): {(1, 6, 9)T , (0, 1, 8)T , (0, 0, 1)T }. 1 0 0 0 (b) U can be reduced to R = 0 1 0 −1 , 0 0 1 2 which means that (0, 1, −2, 1) is a special solution. Basis of N (A): {(0, 1, −2, 1)}. (c) Basis of C(AT ): {(1, 2, 3, 4), (0, 1, 2, 3), (0, 0, 1, 2)}. (d) Basis of N (AT ): none 6. (20 points) Find an orthonormal set q1 , q2 , q3 for which q1 , q2 span the column space of 1 1 A = 2 −1 . −2 4 What fundamental subspace contains q3 ? What is the least-squares solution of Ax = b if b = [1 2 7]T ? Solution: 1 1/3 2 2/3 (a) q1 = a1 /ka1 k = √ 2 12 = , 1 +2 +(−2)2 −2 −2/3 1 1/3 2 B = a2 − (aT2 · q1 )q1 = −1 + 3 2/3 = 1 4 −2/3 2 2/3 q2 = B/kBk = 1/3 2/3 0 1/3 2/3 −2 T T C = a3 − (a3 · q1 )q1 − (a3 · q2 )q2 = 0 + 6 2/3 − 6 1/3 = 2 9 −2/3 2/3 1 −2/3 q3 = C/kCk = 2/3 1/3 (b) q3 is contained in the left nullspace of A. # # " 1/3 2/3 " 1 1 h i aT · q aT · q 3 −3 1 1 2 1 = 2/3 1/3 = QR (c) A = 2 −1 = q1 q2 0 aT2 · q2 0 3 −2/3 2/3 −2 4 Ax = b, QRx = b, Rˆ x = QT b; then " #" # " # 3 −3 xˆ1 −3 ˆ = (1, 2)T . solving = produces x 0 3 xˆ2 6 7. (15 points) The m errors ei are independent with variance σ 2 , so the average of (b − Ax)(b − Ax)T is σ 2 I. Mutiply on the left by (AT A)−1 AT , on the right by A(AT A)−1 , and show that the average b − x)(x b − x)T is σ 2 (AT A)−1 , which is the covariance matrix for the error in x b. of (x Solution: E((b − Ax)(b − Ax)T ) ((AT A)−1 AT )E((b − Ax)(b − Ax)T )(A(AT A)−1 ) E((AT A)−1 AT (b − Ax)(b − Ax)T A(AT A)−1 ) E(((AT A)−1 AT b − (AT A)−1 AT Ax)(bT − xT AT )A(AT A)−1 ) b − x)(bT A(AT A)−1 − xT AT A(AT A)−1 )) E((x b − x)(((AT A)−1 AT b)T − xT )) E((x b − x)(x b T − xT )) E((x b − x)(x b − x)T ) E((x = = = = = = = = σ2I ((AT A)−1 AT )(σ 2 I)(A(AT A)−1 ) σ 2 (AT A)−1 AT A(AT A)−1 σ 2 (AT A)−1 σ 2 (AT A)−1 σ 2 (AT A)−1 σ 2 (AT A)−1 σ 2 (AT A)−1

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