Linear Algebra with Applications Ninth Edition Steven J Leon University of Massachusetts, Dartmouth Boston Columbus Indianapolis New York San Francisco Amsterdam Cape Town Dubai London Madrid Milan Munich Paris Montreal Toronto Delhi Mexico City Sao Paulo Sydney Hong Kong Seoul Singapore Taipei Tokyo Editor in Chief: Deirdre Lynch Acquisitions Editor: William Hoffman Editorial Assistant: Salena Casha Executive Marketing Manager: Jeff Weidenaar Marketing Assistant: Brooke Smith Project Management Team Lead: Christina Lepre Production Project Manager: Mary Sanger Procurement Manager: Carol Melville Senior Author Support/Technology Specialist: Joe Vetere Cover Designer: Heather Scott Image Manager: Diahanne Lucas Cover Image: Ola Dusegard/E+/Getty Images Production Coordination, Composition, Illustrations: Integra Printer/Binder: Edwards Brothers Cover Printer: Phoenix Color Copyright © 2015, 2010, 2006 by Pearson Education, Inc., or its affiliates All Rights Reserved Printed in the 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licensees or distributors Library of Congress Cataloging-in-Publication Data Leon, Steven J Linear algebra with applications / Steven J Leon, University of Massachusetts, Dartmouth – Ninth edition p cm Includes bibliographical references ISBN-13: 978-0-321-96221-8 (hardcover) ISBN-10: 0-321-96221-4 (hardcover) Algebras, Linear–Textbooks I Title QA184.2.L46 2015 512’.5–dc23 2013037914 10 —EBM—18 17 16 15 14 www.pearsonhighered.com ISBN-10: 0-321-96221-4 ISBN-13: 978-0-321-96221-8 To the memories of Florence and Rudolph Leon, devoted and loving parents and to the memories of Gene Golub, Germund Dahlquist, and Jim Wilkinson, friends, mentors, and role models This page intentionally left blank Contents Preface Matrices and Systems of Equations 1.1 1.2 1.3 1.4 1.5 1.6 Systems of Linear Equations Row Echelon Form Matrix Arithmetic Matrix Algebra Elementary Matrices Partitioned Matrices MATLAB Exercises Chapter Test A—True or False Chapter Test B Determinants 2.1 2.2 2.3 ix The Determinant of a Matrix Properties of Determinants Additional Topics and Applications MATLAB Exercises Chapter Test A—True or False Chapter Test B 1 11 27 46 60 70 80 84 85 87 87 94 101 109 111 111 Vector Spaces 112 3.1 3.2 3.3 3.4 3.5 3.6 112 119 130 141 147 157 165 166 167 Definition and Examples Subspaces Linear Independence Basis and Dimension Change of Basis Row Space and Column Space MATLAB Exercises Chapter Test A—True or False Chapter Test B v vi Contents Linear Transformations 169 4.1 4.2 4.3 169 178 192 198 199 200 Orthogonality 5.1 5.2 5.3 5.4 5.5 5.6 5.7 Definition and Examples Matrix Representations of Linear Transformations Similarity MATLAB Exercises Chapter Test A—True or False Chapter Test B The Scalar Product in Rn Orthogonal Subspaces Least Squares Problems Inner Product Spaces Orthonormal Sets The Gram–Schmidt Orthogonalization Process Orthogonal Polynomials MATLAB Exercises Chapter Test A—True or False Chapter Test B 201 202 217 225 238 247 266 275 283 285 285 Eigenvalues 287 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 288 301 312 330 342 356 370 377 387 393 393 Eigenvalues and Eigenvectors Systems of Linear Differential Equations Diagonalization Hermitian Matrices The Singular Value Decomposition Quadratic Forms Positive Definite Matrices Nonnegative Matrices MATLAB Exercises Chapter Test A—True or False Chapter Test B Numerical Linear Algebra 395 7.1 7.2 396 404 Floating-Point Numbers Gaussian Elimination Contents 7.3 7.4 7.5 7.6 7.7 Iterative Methods 8.1 Pivoting Strategies Matrix Norms and Condition Numbers Orthogonal Transformations The Eigenvalue Problem Least Squares Problems MATLAB Exercises Chapter Test A—True or False Chapter Test B Online∗ Online∗ Nilpotent Operators The Jordan Canonical Form Appendix: MATLAB Bibliography Answers to Selected Exercises Index ∗ 409 415 429 440 451 463 468 468 Basic Iterative Methods Canonical Forms 9.1 9.2 vii 471 483 486 499 Online: The supplemental Chapters and can be downloaded from the Internet See the section of the Preface on supplementary materials This page intentionally left blank 490 Answers to Selected Exercises ⎧ ⎪ ⎪ ⎪ (a) ⎪ ⎪ ⎩ − 12 − 12 ⎧ ⎩2 (c) ⎪ ⎫ ⎪ ⎪ ⎪ ⎪; ⎪ ⎭ ⎧ ⎩ 11 (b) ⎪ −4 ⎫ 14 ⎪ ⎭; −5 ⎫ 3⎪ ⎭ 4 [x]E = (−1, 2)T, [y]E = (5, −8)T, [z]E = (−1, 5)T ⎧ ⎫ −1 ⎪ ⎪ ⎪ ⎪ ⎪ T ⎪ −1 ⎪ (a) ⎪ ⎪ −1 ⎪; (b) (1, −4, 3) ; ⎩ ⎭ −1 (c) (0, −1, 1)T; (d) (2, 2, −1)T ⎧ ⎫ ⎧ ⎫ −1 −2 ⎪ 7⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 5⎪ ⎪ 1 0⎪ (a) ⎪ ⎪ ⎪ ⎪ ⎩ ⎭; (b) ⎪ ⎩ ⎭ 1 −2 10 11 17 w1 = (5, 9)T and w2 = (1, 4)T u1 = (0, −1)T and u2 = (1, 5)T ⎧ ⎧ ⎫ ⎪ −1 2 ⎪ ⎪ ⎪ ⎭; (b) ⎪ (a) ⎩ ⎪ ⎩1 −1 ⎧ ⎫ 0⎪ ⎪ ⎪ −1 ⎪ ⎪0 ⎪ −1 ⎪ 10 ⎪ ⎪ ⎪ ⎩ ⎭ 0 3.6 (a) 3; (b) 3; ⎫ ⎪ ⎪ ⎪ ⎪ ⎭ 19 23 (c) (a) u2 , u4 , u5 are the column vectors of U corresponding to the free variables u2 = 2u1 , u4 = 5u1 − u3 , u5 = −3u1 + 2u3 (a) consistent; 18 4.2 (b) inconsistent; (e) consistent (a) infinitely many solutions; (c) unique solution rank of A = 3; dim N(B) = 1; 18 (b) n − 32 If xj is a solution to Ax = ej for j = 1, , m and X = (x1 , x2 , , xm ), then AX = Im CHAPTER TEST A True False False False True True False True True 10 False 11 True 12 False 13 True 14 False 15 False Chapter 4.1 (a) reflection about x2 axis; (b) reflection about the origin; (c) reflection about the line x2 = x1 ; (d) the length of the vector is halved; (e) projection onto x2 axis (7, 18)T All except (c) are linear transformations from R3 into R2 (b) and (c) are linear transformations from R2 into R3 (a), (b), and (d) are linear transformations (a) and (c) are linear transformations from P2 into P3 L(ex ) = ex − and L(x2 ) = x3 /3 (a) and (c) are linear transformations from C[0, 1] into R1 (a) ker(L) = {0}, L(R3 ) = R3 ; (c) ker(L) = Span(e2 , e3 ), L(R3 ) = Span((1, 1, 1)T ) (a) L(S) = Span(e2 , e3 ); (b) L(S) = Span(e1 , e2 ) (a) ker(L) = P1 , L(P3 ) = Span(x2 , x); (c) ker(L) = Span(x2 − x), L(P3 ) = P2 The operator in part (a) is one-to-one and onto ⎧ ⎫ ⎧ ⎫ ⎩ −1 ⎪ ⎭; (c) ⎪ ⎩0 1⎪ ⎭; (a) ⎪ 1 ⎧ ⎫ ⎧ ⎫ ⎪ 0⎪ ⎪ 0⎪ ⎪ ⎪ ⎪ ⎩ ⎭ (d) ⎪ ; (e) ⎪ ⎪ ⎩0 ⎭ ⎧ ⎫ ⎧ ⎫ ⎩1 0⎪ ⎭; (b) ⎪ ⎩1 0⎪ ⎭; (a) ⎪ 0 0 ⎧ ⎫ 0⎪ ⎩ −1 ⎭ (c) ⎪ −1 ⎧ ⎫ ⎧ ⎫ 0 1⎪ 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪0 0⎪ ⎪; (b) ⎪ ⎪1 0⎪ ⎪; (a) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 0 1 ⎧ ⎫ 0 2⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0⎪ (c) ⎪ ⎪ ⎪ ⎩ ⎭ −1 (a) (0, 0, 0)T; (b) (2, −1, −1)T; (c) (−15, 9, 6)T ⎧ ⎫ 1 ⎪ ⎪ ⎪ √ √ ⎪ ⎧ ⎫ ⎪ ⎪ ⎪ 1⎪ ⎪ ⎪ 2⎪ ⎪ ⎪ ⎪ ⎩ ⎭; (a) ⎪ ; (b) ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎩ −√ ⎭ √ ⎪ 2 ⎫ ⎧√ ⎧ ⎫ −1 ⎪ ⎪ √ ⎪ ⎩0 1⎪ ⎭ (c) ⎪ ⎭; (d) ⎪ ⎩ 0 Answers to Selected Exercises ⎧ ⎪ ⎪ ⎪0 ⎪ ⎪ ⎩ 10 13 14 15 18 4.3 ⎫ 0⎪ ⎪ ⎪ 1⎪ ; ⎪ ⎭ ⎧ ⎫ 0 1⎪ ⎪ ⎪ ⎪ ⎪0 ⎪ −1 ⎪ (b) ⎪ ⎪ ⎪ ⎩ ⎭ −1 ⎧ ⎫ 1 1⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1⎪ (a) ⎪ ; ⎪ ⎪ ⎩ ⎭ −2 −1 (b) (i) 7y1 + 6y2 − 8y3 , (ii) 3y1 + 3y2 − 3y3 , (iii) y1 + 5y2 + 3y3 (a) square; (b) (i) contraction by a factor 12 , (ii) clockwise rotation by 45◦ , (iii) translation units to the right and units down √ ⎧ ⎫ − − 23 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎪ ⎪ ⎪ (a) ⎪ ⎪; − ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ 0 ⎧ ⎫ ⎧ ⎫ −3 ⎪ −1 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪0 ⎪; (d) ⎪ ⎪ 2⎪ ⎪ ⎪ 5⎪ (b) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 0 0 ⎫ ⎧ ⎪ 12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪; ⎪ ⎪ ⎪ ⎭ ⎩ ⎫ ⎫ ⎧ ⎧ ⎪ ⎧ ⎫ ⎪ ⎪ 12 12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪; (a) ⎪ ⎪ (d) ⎪ ⎭ ⎩ 5⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎭ ⎩ ⎩ −8 −2 0 −2 ⎧ ⎫ 1 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪0 2⎪ ⎪; ⎪ ⎪ ⎩ ⎭ 0 ⎫ ⎫ ⎧ ⎧ ⎭ ⎭; (c) ⎪ ⎩ −2 −4 ⎪ ⎩ −1 −3 ⎪ (a) ⎪ −1 3 For the matrix A, see the answers to Exercise of Section 4.2 ⎧ ⎫ ⎧ ⎫ 0⎪ ⎩0 1⎪ ⎭; (b) B = ⎪ ⎩ −1 ⎭; (a) B = ⎪ 0 −1 ⎫ ⎧ ⎫ ⎧ ⎪ ⎪ ⎪ 0⎪ 0⎪ ⎪ ⎪ ⎪ ⎪ ⎭; (d) B = ⎪ (c) B = ⎩ ⎪; ⎪ ⎩ −1 ⎭ ⎫ ⎧ 1 ⎪ ⎪ ⎪2 2⎪ ⎪ ⎪ ⎪ (e) B = ⎪ ⎪ ⎭ ⎩1 1⎪ 2 ⎫ ⎧ 1⎪ ⎭; (b) ⎪ ⎩ −4 −3 ⎧ ⎫ −1 −1 ⎪ ⎪ ⎪ ⎪ ⎪ −1 ⎪ −1 ⎪ B = A = ⎪ ⎪ ⎪ ⎩ ⎭ −1 −1 ⎧ ⎩ (a) ⎪ −1 ⎫ 0⎪ ⎭ −1 491 (Note: in this case the matrices A and U commute; so B = U −1 AU = U −1 UA = A.) ⎧ ⎫ ⎧ ⎫ 1 0⎪ 0 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪1 ⎪, B = ⎪ ⎪0 0⎪ ⎪ ⎪ −2 ⎪ V = ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 1 0 ⎧ ⎫ ⎧ ⎫ 0 2⎪ 0 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0⎪ 0⎪ (a) ⎪ ; (b) ⎪ ; ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 0 0 ⎧ ⎫ 1⎪ ⎪ ⎪ ⎪0 0⎪ ⎪; (d) a x + a 2n (1 + x2 ) ⎪ (c) ⎪ ⎪ ⎪ ⎩ ⎭ 0 ⎧ ⎫ ⎧ ⎫ 0⎪ 0 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪0 ⎪; (b) ⎪ ⎪0 1⎪ ⎪; 1⎪ (a) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ −1 ⎧ ⎫ 0 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0⎪ (c) ⎪ ⎪ ⎪ ⎩ ⎭ 0 −1 CHAPTER TEST A False True True False False True True True True 10 False Chapter 5.1 (a) 0◦ ; (b) 90◦ √ (a) 14 (scalar projection), (2, 1, 3)T (vector projection); (b) 0, 0; √ (d) 21 21 T (c) √ T 14 13 , 13 ( 42 , 28 )T ; 13 13 16 32 T , ( 21 , 21 , 21 ) (a) p = (3, 0)T, x − p = (0, 4)T, pT (x − p) = · + · = 0; (c) p = (3, 3, 3)T, x − p = (−1, 1, 0)T, pT (x − p) = −1 · + · + · = (1.8, 3.6) (1.4, 3.8) 0.4 (a) 2x + 4y + 3z = 0; 10 (c) z − = 20 The correlation matrix with entries rounded to two decimal places is ⎧ 1.00 ⎪ ⎪ ⎪ ⎪ −0.04 ⎪ ⎩ 0.41 −0.04 1.00 0.87 ⎫ 0.41 ⎪ ⎪ ⎪ 0.87 ⎪ ⎪ ⎭ 1.00 492 Answers to Selected Exercises 5.2 (a) {(3, 4)T } basis for R(AT ), {(−4, 3)T } basis for N(A), {(1, 2)T } basis for R(A), {(−2, 1)T } basis for N(AT ); 16 5.5 (1a) p = (3, 1, 0)T, r = (0, 0, 2)T (1c) p = (3.4, 0.2, 0.6, 2.8)T, r = (0.6, −0.2, 0.4, −0.8)T (a) {(1 − 2α, α)T | α real}; (b) {(2 − 2α, − α, α)T | α real} (a) p = (1, 2, −1)T, b − p = (2, 0, 2)T ; (b) p = (3, 1, 4)T, p − b = (−5, −1, 4)T (a) y = 1.8 + 2.9x 0.55 + 1.65x + 1.25x2 14 The least squares circle will have center (0.58, −0.64) and radius 2.73 (answers rounded to two decimal places) 5.6 (b) r = (0.2605, 0.2337, 0.2850, 0.2208) √ x = 2, y = 6, x + y = 10 (a) 0; (b) 5; (a) 1; (b) (c) 7; 11 (a) √ ⎧ ⎪ ⎪ ⎪ ⎪ (b) ⎪ ⎪ ⎪ ⎩ √ 74 15 (a) x = 7, x (b) x = 4, x (c) x = 3, x (b) p = 32 x √ (b) 34 = 5, x ∞ = 4; √ = 6, x ∞ = 2; √ = 3, x ∞ = + √ 1/2 = π π √1 √2 √1 T T , √1 , √1 2 , − √15 , √25 T ; T ⎫ √ ⎪ ⎪⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎭ √ ⎫ ⎪ √2 ⎪ ⎭; ⎫ − √15 ⎪ ⎧√ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ ⎭ √ ⎫ 4√5 ⎪ ⎪ ⎭ ⎧ − √2 ⎪ ⎪ ⎪ ⎪ (a) ⎪ ⎪ ⎪ ⎩ 1 π π ; √ 10 ; (a) (d) √ u = 3, v = 2; (c) √2 , √1 5 (b) T p= + 53 u2 , √ − 32 − √12 , (a) 15 (a) w = (0.1995, 0.2599, 0.3412, 0.1995)T (a) θ = =2 21 (b) (i) (2, −2)T , (ii) (5, 2)T, (iii) (3, 1)T ⎫ ⎧1 0⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 22 (a) P = ⎪ ⎪ ⎪ ⎪; 1 ⎪ ⎪ ⎪ ⎪ 0 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 ⎭ 0 2 ⎫ ⎧ − 12 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ − 0 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ 23 (b) Q = ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ ⎪ 0 − ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 ⎭ 0 −2 √ √ 29 (b) = 2, x = 36 ; (c) l(x) = 97 x (c) (1.6, 0.6, 1.2)T (b) x = 1, y = u (b) (i) 0, (ii) − π2, (iii) 0, (iv) (b) 8x − 2y + z = ( 43 , 13 , 13 , 0)T √ (a) 15; (b) 10 dim N(A) = n − r, dim N(AT ) = m − r π ; ∞ (c) norm (b) c1 = y1 cos θ + y2 sin θ, c2 = −y1 sin θ + y2 cos θ {(−1, 2, 0, 1)T , (2, −3, 1, 0)T } is one basis for S⊥ 5.4 = 3, x − y 5 , 41 , )T, p − x = ( 18 , 18 , − 10 )T p = ( 23 18 18 9 (b) The orthogonal complement is spanned by (−5, 1, 3)T (a) (2, 1)T ; (b) norm; (a) and (d) x = (a) {(1, 1, 0)T , (−1, 0, 1)T } 5.3 = 5, x − y (b) x = − (d) basis for R(AT ): {(1, 0, 0, 0)T , (0, 1, 0, 0)T (0, 0, 1, 1)T }, basis for N(A): {(0, 0, −1, 1)T }, basis for R(A): {(1, 0, 0, 1)T , (0, 1, 0, 1)T (0, 0, 1, 1)T }, basis for N(AT ): {(1, 1, 1, −1)T } (a) N = (8, −2, 1)T ; x−y 28 (a) not a norm; √1 √1 √ ( 13 , 23 , − 23 )T , ( 23 , 13 , 23 )T , (− 23 , 23 , 13 )T u1 (x) = u3 (x) = (a) √1 , √ 10 u2 (x) = x2 − (2, 1, 2)T , √ x, √ (−1, 4, −1)T ; Answers to Selected Exercises ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (b) Q = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ √ − √ 2 3 √ − ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎧ ⎪ ⎪3 ⎪ R=⎪ ⎪ ⎩ ⎧ ⎫ 9⎭ ⎪ ⎪ ⎩ (c) x = −3 ⎫ ⎧3 − 5√4 ⎪ ⎪ ⎪ ⎪ ⎪⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ √ ⎪ (b) ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ √1 √ (b) λ1 = 3, the eigenspace is spanned by (4, 3)T , λ2 = 2, the eigenspace is spanned by (1, 1)T ; ⎫ ⎪ ⎪ ⎪ ⎪ ; ⎪ ⎭ (c) λ1 = λ2 = 2, the eigenspace is spanned by (1, 1)T , (d) λ1 = + 4i, the eigenspace is spanned by (2i, 1)T , λ2 = − 4i, the eigenspace is spanned by (−2i, 1)T ; ⎫ ⎪ ⎪ √ ⎭; 2 (e) λ1 = + i, the eigenspace is spanned by (1, + i)T , λ2 = − i, the eigenspace is spanned by (1, − i)T ; (f) λ1 = λ2 = λ3 = 0, the eigenspace is spanned by (1, 0, 0)T ; (c) (2.1, 5.5)T − √12 , √12 , 0, ⎧⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎩ ⎩⎪ 5.7 5 5 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − 25 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − 25 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ √ T , ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ √ √ √ , 32 , − 22 , 62 T (g) λ1 = 2, the eigenspace is spanned by (1, 1, 0)T , λ2 = 1, the eigenspace is spanned by (1, 0, 0)T , (0, 1, −1)T ; ⎫⎫ ⎧ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − √12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭⎪ ⎩ ⎭ (h) λ1 = 1, the eigenspace is spanned by (1, 0, 0)T , λ2 = 4, the eigenspace is spanned by (1, 1, 1)T , λ3 = −2, the eigenspace is spanned by (−1, −1, 5)T ; (i) λ1 = 2, the eigenspace is spanned by (7, 3, 1)T , λ2 = 1, the eigenspace is spanned by (3, 2, 1)T , λ3 = 0, the eigenspace is spanned by (1, 1, 1)T ; (a) T4 = 8x4 −8x2 +1, T5 = 16x5 −20x3 +5x; (b) H4 = 16x4 − 48x2 + 12, H5 = 32x5 − 160x3 + 120x p1 (x) = x, p2 (x) = x2 − π (j) λ1 = λ2 = λ3 = −1, the eigenspace is spanned by (1, 0, 1)T ; +1 p(x) = (sinh 1)P0 (x) + 3e P1 (x) + sinh − 3e P2 (x) p(x) ≈ 0.9963 + 1.1036x + 0.5367x2 (k) λ1 = λ2 = 2, the eigenspace is spanned by e1 and e2 , λ3 = 3, the eigenspace is spanned by e3 , λ4 = 4, the eigenspace is spanned by e4 ; (a) U0 = 1, U1 = 2xU2 = 4x2 − 11 p(x) = (x − 2)(x − 3) + (x − 1)(x − 3) + 2(x − 1)(x − 2) 13 · f − √13 + · f (l) λ1 = 3, the eigenspace is spanned by (1, 2, 0, 0)T , λ2 = 1, the eigenspace is spanned by (0, 1, 0, 0)T , λ3 = λ4 = 2, the eigenspace is spanned by (0, 0, 1, 0)T √1 14 (a) degree or less; (b) the formula gives the exact answer for the first integral The approximate value for the second integral is 1.5, π while the exact answer is CHAPTER TEST A False False False False True False True True True 10 False Chapter 6.1 493 (a) λ1 = 5, the eigenspace is spanned by (1, 1)T , λ2 = −1, the eigenspace is spanned by (1, −2)T ; 10 β is an eigenvalue of B if and only if β = λ − α for some eigenvalue λ of A 14 λ1 = 6, λ2 = 2; 6.2 24 λ1 xT y = (Ax)T y = xT AT y = λ2 xT y ⎧ ⎫ ⎪ c1 e2t + c2 e3t ⎪ ⎪ ⎪ ⎪ (a) ⎪ ⎩ ⎭; c1 e2t + 2c2 e3t ⎫ ⎧ ⎪ −c1 e−2t − 4c2 et ⎪ ⎪ ⎪ ⎪; ⎪ (b) ⎩ ⎭ c1 e−2t + c2 et ⎫ ⎧ ⎪ 2c1 + c2 e5t ⎪ ⎪; ⎪ ⎪ ⎪ (c) ⎩ ⎭ c1 − 2c2 e5t 494 Answers to Selected Exercises ⎫ ⎧ ⎪ −c1 et sin t + c2 et cos t ⎪ ⎪ ⎪ ⎪ (d) ⎪ ⎩ ⎭; c1 et cos t + c2 et sin t ⎧ ⎪ −c1 e3t sin 2t + c2 e3t cos 2t ⎪ (e) ⎪ ⎩ c1 e3t cos 2t + c2 e3t sin 2t ⎫ ⎧ −c1 + c2 e5t + c3 et ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −3c + 8c e5t ⎪ (f) ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 5t c1 + 4c2 e ⎫ ⎧ −3t t ⎪ e + 2e ⎪ ⎪ (a) ⎪ ⎭; ⎩ −3t −e + 2et ⎧ t ⎫ e cos 2t + 2et sin 2t ⎪ ⎪ ⎪ ⎪ (b) ⎩ t ⎭; e sin 2t − 2et cos 2t ⎫ ⎧ −6et + 2e−t + ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪; ⎪ ⎪ (c) ⎪ −3et + e−t + ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ t −t −e + e + ⎫ ⎧ −2 − 3et + 6e2t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ t 2t ⎪ ⎪ ⎪ (d) ⎪ + 3e − 3e ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 2t + 3e In the long run we would expect 60 percent of the employees to be enrolled ⎧ ⎫ 0.70 0.20 0.10 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0.20 0.70 0.10 ⎪ 22 (a) A = ⎪ ⎪ ⎪ ⎩ ⎭ 0.10 0.10 0.80 (c) The membership of all three groups will approach 100,000 as n gets large ⎫ ⎪ ⎪ ⎪ ⎭; y1 (t) = 15e−0.24t + 25e−0.08t, y2 (t) = −30e−0.24t + 50e−0.08t ⎧ √ √ ⎪ −2c1 et − 2c2 e−t + c3 e 2t + c4 e− 2t ⎪ ⎪ (a) ⎪ √ √ ⎪ ⎩ c1 et + c2 e−t − c3 e 2t − c4 e− 2t ⎫ ⎧ ⎪ c1 e2t + c2 e−2t − c3 et − c4 e−t ⎪ ⎪ ⎪ ⎪ ⎪ (b) ⎩ ⎭ c1 e2t − c2 e−2t + c3 et − c4 e−t 26 The transition matrix is ⎧ 12 ⎪ ⎪ ⎪ ⎪1 ⎪ ⎪ ⎪3 A = 0.85 ⎪ 1 ⎪ ⎪ ⎪3 ⎪ ⎩1 ⎧1 ⎪ ⎪ ⎪ 14 ⎪ ⎪ ⎪ ⎪4 + 0.15 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 14 y1 (t) = −e2t + e−2t + et ; y2 (t) = −e2t − e−2t + 2et √ x1 (t) = cos t + sin t + √13 sin 3t, √ x2 (t) = cos t + sin t − √13 sin 3t m2 x2 (t) = −k(x2 − x1 ) + k(x3 − x2 ) m3 x3 (t) = −k(x3 − x2 ) − kx3 √ ⎫ √ ⎧ 0.1 cos 3t + 0.9 cos 2t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √ ⎪ √ ⎪ ⎪ (b) ⎪ ⎪ −0.2 cos 3t + 1.2 cos 2t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √ ⎪ √ ⎩ ⎭ 0.1 cos 3t + 0.9 cos 2t 6.3 11 p(λ) = (−1)n (λn − an−1 λn−1 − · · · − a1 λ − a0 ) (b) α = 2; (c) α = or α = −1; (d) α = 1; (e) α = 0; (g) all values of α 21 The transition matrix and steady-state vector for the Markov chain are ⎫ ⎫ ⎧ ⎧ ⎪ ⎭ ⎩ 0.60 ⎪ ⎭ x=⎪ ⎩ 0.80 0.30 ⎪ 0.40 0.20 0.70 0 1 4 4 4 4 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ 4 4 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎫ ⎧ ⎭ ⎩e e⎪ 30 (b) ⎪ e ⎧ ⎫ ⎪ − 2e 1− e ⎪ ⎪ ⎪ ⎪ 31 (a) ⎪ ⎩ ⎭; −6 + 6e −2 + 3e ⎧ ⎫ e −1 + e −1 + e ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (c) ⎪ − e − e − e ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ −1 + e −1 + e e ⎧ ⎫ ⎧ −t ⎫ t − e−t ⎪ ⎭; (b) ⎪ ⎩ −3e ⎭; ⎩ e−t ⎪ 32 (a) ⎪ e et + e−t ⎧ t ⎫ 3e − ⎪ ⎪ ⎪ ⎪ ⎪ −t ⎪ 2−e ⎪ (c) ⎪ ⎪ ⎪ ⎭ ⎩ e−t ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ 10 (a) m1 x1 (t) = −kx1 + k(x2 − x1 ) 4 4 6.4 (a) z = 6, w = 3, z, w = −4 + 4i, w, z = −4 − 4i; z = 4, w = 7, z, w = −4 + 10i, w, z = −4 − 10i √ (b) z = 4z1 + 2z2 (b) (a) uH1 z = + 2i, zH u1 = − 2i, uH2 z = − 5i, zH u2 = + 5i; (b) z =9 (b) and (f) are Hermitian while (b), (c), (e), and (f) are normal 14 (b) Ux = (Ux)H Ux = xH U H Ux = xH x = x 15 U is unitary, since U H U = (I − 2uuH )2 = I − 4uuH + 4u(uH u)uH = I Answers to Selected Exercises 24 λ1 = 1, λ2 = −1, u1 = √1 , √1 2 ⎧ ⎪ ⎪ ⎪ A = 1⎪ ⎪ ⎩ 6.5 2 2 √ T (b) det(A1 ) = 3, det(A2 ) = −10, not positive definite; (c) det(A1 ) = 6, det(A2 ) = 14, det(A3 ) = −38, not positive definite; (d) det(A1 ) = 4, det(A2 ) = 8, det(A3 ) = 13, positive definite (2) a11 = 3, a(1) 22 = 2, a33 = ⎧ ⎫⎧ ⎫ ⎫⎧ 0⎪⎪ 12 ⎪ ⎪1 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a) ⎪ ⎩ ⎭ ⎩1 ⎭ ⎩ ⎭; 1 ⎫ ⎫⎧ ⎧ ⎫⎧ 0⎪ 0⎪⎪ − 13 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (b) ⎩ ⎭⎩ ⎭; ⎭⎩ −3 1 ⎫⎧ ⎧ ⎫⎧ ⎫ 16 0 ⎪ ⎪ 12 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎪ ⎪ 12 0⎪ ⎪ ⎪ ⎪ ⎪ 0⎪ −1 ⎪ (c) ⎪ ; ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩ ⎪ ⎭⎩ ⎭ ⎭ ⎩ T , u2 = − √12 , √12 , ⎫ ⎫ ⎧ ⎪ ⎪ − 12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + (−1) ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 1 ⎭ −2 σ1 = 10, σ2 = 0; σ1 = 3, σ2 = 2; σ1 = 4, σ2 = 2; σ1 = 3, σ2 = 2, σ3 = The matrices U and V are not unique The reader may check his or her answers by multiplying out U V T ⎫ ⎧ ⎪ ⎪ 1.2 −2.4 ⎭ (b) rank of A = 2, A = ⎩ −0.6 1.2 The closest matrix of rank is ⎧ ⎫ −2 20 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 14 19 10 ⎪ ⎪ ⎪ ⎩ ⎭, 0 (a) (b) (c) (d) −3 (a) (b) (a) basis for R(AT ): {v1 = ( 23 , 23 , 13 )T , v2 = (− 23 , 13 , 23 )T }; 6.6 ⎫ ⎪ ⎪ ⎪ ⎪; 1⎭ ⎧ ⎪ ⎪ (a) ⎪ ⎪ ⎩ −5 − 52 ⎧ ⎩1 (a) Q = √12 ⎪ ellipse; ⎧ ⎩ (d) Q = √12 ⎪ −1 √ √ = − 22 (x − 2) or √ x , parabola (y )2 = − positive definite; (b) indefinite; negative definite; (e) indefinite minimum; (b) saddle point; saddle point; (f) local maximum det(A1 ) = 2, det(A2 ) = 3, positive definite; y + 6.7 (a) (d) (a) (c) (a) ⎫ 1⎪ ⎭, √ 2 (c) ( 13 , − 23 , 23 )T } ⎧ ⎫ 12 −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (b) ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎩ ⎭ −1 ⎫ 2 1⎪ ⎭, (x4) + (y12) = 1, −1 −1 0 0 ⎫⎧ ⎧ ⎫⎧ ⎫ 0 ⎪ ⎪ 13 − 23 ⎪ 0⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0⎪ 1⎪ (d) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭⎩ ⎭ ⎭⎩ ⎩ The closest matrix of rank is ⎧ ⎫ 12 12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 16 16 ⎪ ⎪ ⎪ ⎩ ⎭ 0 basis for N(A): {v3 = 495 (d) 6.8 0 0 λ1 = 4, λ2 = −1, x1 = (3, 2)T ; λ1 = 8, λ2 = 3, x1 = (1, 2)T ; λ1 = 7, λ2 = 2, λ3 = 0, x1 = (1, 1, 1)T λ1 = 3, λ2 = −1, x1 = (3, 1)T ; λ1 = = exp(0), λ2 = −2 = exp(π i), x1 = (1, 1)T ; (c) λ1 = = exp(0), √ , λ2 = −1 + 3i = exp 2πi √ λ3 = −1 − 3i = exp 4πi , x1 = (4, 2, 1)T x1 = 70,000, x2 = 56,000, x3 = 44,000 x1 = x2 = x3 (I − A)−1 = I + A + · · · + Am−1 ⎧ ⎫ −1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0 1⎪ (a) (I − A)−1 = ⎪ ⎪ ⎪ ⎩ ⎭; −1 (a) (b) (c) (a) (b) 1 ⎧ ⎫⎧ ⎫ ⎪ ⎩2 0⎪ ⎭⎪ ⎩2 1⎪ ⎭; 3 ⎧ ⎫⎧ ⎫ ⎪ ⎩ 0⎪ ⎭⎪ ⎩ −1 ⎪ ⎭; −1 1 ⎫⎧ ⎧ ⎫ 0⎪ √2 1⎪ ⎪ ⎪ √ ⎪ ⎪ √ ⎪ ⎪ ⎪⎪ ⎪ ⎪ ⎪ ⎪ ⎪0 2 − 2⎪ ⎪ ⎪ ⎪ ⎪ √2 ⎪ ⎭; ⎩ ⎭⎪ ⎩ 0 − 2 ⎫⎧ ⎫ ⎧ √0 0⎪ −2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √1 √ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √3 ⎪ √3 √0 ⎪ ⎪ ⎪ ⎪ ⎪0 ⎪ ⎪ ⎪ ⎩ ⎭⎪ ⎭ ⎩ 2 −2 0 496 Answers to Selected Exercises ⎧ ⎪ ⎪ ⎪0 (b) A2 = ⎪ ⎪ ⎩ ⎧ ⎪ ⎪0 ⎪ A3 = ⎪ ⎪ ⎩ ⎫ −2 ⎪ ⎪ ⎪ 0⎪ , ⎪ ⎭ 0 ⎫ 0⎪ ⎪ ⎪ 0⎪ ⎪ ⎭ 0 (b) (i) 156 multiplications and 105 additions, (ii) 47 multiplications and 24 additions, (iii) 100 multiplications and 60 additions 5n − multiplications/divisions, 3n − additions/subtractions (a) [(n − j)(n − j + 1)]/2 multiplications; [(n − j − 1)(n − j)]/2 additions; (b) and (c) are reducible , 12 , , )T 15 (d) w = ( 12 29 29 29 29 ≈ (0.4138, 0.4138, 0.1034, 0.0690)T CHAPTER TEST A True False True False False False False False True 10 False 11 True 12 True 13 True 14 False 15 True Chapter 7.1 (a) 0.231 × 10 ; (b) 0.326 × 10 ; (c) 0.128 × 10−1 ; (a) (d) 0.824 × 105 −4 = −2; δ ≈ −8.7 × 10 ; (b) = 0.04; δ ≈ 1.2 × 10−3 ; (c) = 3.0 × 10−5 ; δ ≈ 2.3 × 10−3 ; (d) = −31; δ ≈ −3.8 × 10−4 (a) (1.0101)2 × 24 ; (c) (1.0100)2 × 23 ; (a) 10,420, (b) 0, (b) (1.1000)2 × 2−2 ; (d) −(1.1010)2 × 2−4 = −0.0018, δ ≈ −1.7 × 10−7 ; = −8, δ = −1; (c) × 10−4 , (d) 82,190, = × 10−5 , δ = 1; = 25.7504, δ ≈ 3.1 × 10−4 (a) 0.1043 × 106 ; (b) 0.1045 × 106 ; (c) 0.1045 × 106 23 7.2 (a) (1.00111000000000000000000)2 × 23 or 9.75 ⎧ ⎫⎧ ⎫ 0⎪ 1 1⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −1 ⎪ ⎪ 0⎪ A = ⎪ ⎪ ⎪ ⎪ ⎩ ⎭⎪ ⎩ ⎭ −3 0 (a) (2, −1, 3)T ; (b) (1, −1, 3)T ; (c) (1, 5, 1)T (a) n2 multiplications and n(n − 1) additions; (b) n3 multiplications and n2 (n−1) additions; (c) (AB)x requires n + n multiplications and n3 − n additions; A(Bx) requires 2n2 multiplications and 2n(n − 1) additions (c) It requires on the order of 23 n3 additional multiplications/divisions to compute A−1 given the LU factorization 7.3 (a) (1, 1, −2); ⎧ ⎫⎧ ⎫⎧ ⎫ 0 1⎪ 0⎪ −2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪2 0⎪ ⎪ ⎪0 ⎪ 0⎪ 8⎪ (b) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭⎪ ⎩ ⎭⎪ ⎩ ⎭ 0 0 −23 (a) (1, 2, 2); (b) (4, −3, 0); (c) (1, 1, 1) ⎧ ⎪ ⎪0 ⎪ P = ⎪ ⎪ ⎩ ⎧ ⎪ ⎪ ⎪ ⎪ U=⎪ ⎪ ⎪ ⎩ ⎫ 1⎪ ⎪ ⎪ 0⎪ ⎪ ⎭, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ L=⎪ ⎪ ⎪ ⎪ ⎩ 1 − 12 − 13 ⎫ ⎧ 6⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ x=⎪ − ⎪ ⎪ ⎪ ⎭ ⎩ 2⎪ ⎫ −6 ⎪ ⎪ ⎪ ⎪ 9⎪ , ⎪ ⎪ ⎭ ⎧ ⎫ 1⎪ ⎪ ⎪ ⎪ P = Q = ⎩ ⎭, ⎧0 ⎫⎧ ⎪ 0⎪ ⎪ ⎪ ⎪4 ⎪ ⎪ PAQ = LU = ⎪ ⎭⎩ ⎩1 ⎧ ⎫ ⎪ ⎩ ⎭ x=⎪ −2 ⎫ 0⎪ ⎪ ⎪ ⎪ 0⎪ ⎪ , ⎪ ⎪ ⎪ ⎭ ⎫ 2⎪ ⎪ ⎭, (a) cˆ = Pc = (−4, 6)T, y = L−1 cˆ = (−4, 8)T, z = U −1 y = (−3, 4)T (b) x = Qz = (4, −3)T ⎧ ⎫ ⎧ ⎫ 0 1⎪ 0 1⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 0⎪ 0⎪ (b) P = ⎪ , Q = , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭ 0 ⎫ ⎧ ⎧ ⎫ ⎪ 0⎪ ⎪ ⎪ 2⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪, U = ⎪ ⎪0 3⎪ ⎪ ⎪ −1 ⎪ ⎪ ⎪ ⎪ L=⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ ⎪ ⎪ ⎪ ⎭ ⎩ 0 2 −2000e ≈ −3333e If e = 0.001, then 0.6 δ = −3 Error (1.667, 1.001) Answers to Selected Exercises ⎧3 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (a) ⎪ ⎪ ⎪ ⎪ ⎩4 (5.002, 1.000) 7.4 10 (5.001, 1.001) √ (a) A F = 2, A ∞ = 1, A (b) A F = 5, A ∞ (c) A F = A = A (d) A F = 7, A ∞ = 6, A (e) A F = 9, A ∞ = 10, A ∞ = 5, A I = I = 1, I ∞ F √ = = 10; = 12 n; (b) (−1, 1, −1) T (a) 10; 27 (a) Since for any vector y in Rn we have y ∞ ≤ y ≤ √ n y ∞ it follows that Ax ∞ ≤ ≤ Ax A x ≤ √ n A x ∞ 29 cond∞ A = 400 ⎧ ⎫ ⎧ ⎫ ⎩ −0.48 ⎪ ⎭ and⎪ ⎩ −2.902 ⎪ ⎭ 30 The solutions are⎪ 0.8 2.0 31 cond∞ (A) = 28 ⎧ ⎪ ⎩1−n 33 (a) A−1 = n n ⎫ n ⎪ ⎭; −n (b) cond∞ An = 4n; (c) limn→∞ cond∞ An = ∞; 34 σ1 = 8, σ2 = 8, σ3 = 35 (a) r = (−0.06, 0.02)T and the relative residual is 0.012; (b) 20; (d) x = (1, 1)T , x − x ∞ = 0.12 ; 36 cond1 (A) = 37 0.3 38 (a) r ∞ = 0.10, cond∞ (A) = 32; (b) 0.64; 7.5 ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪; ⎪ 0⎪ ⎪ ⎪ ⎭ −5 ⎫ ⎧ √ − √12 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ √ √ (b) ⎪ ; − − ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 0 ⎫ ⎧ 0 ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪0 2 ⎪ ⎪ ⎪ ; (c) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √ ⎭ ⎩ − 12 ⎫ ⎧ 0 ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ − ⎪ 2 ⎪ ⎪ (d) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎭ ⎩ 2 = 6; = 1; 2 = 1; 497 (c) x = (12.50, 4.26, 2.14, 1.10)T, δ = 0.04 ⎫ ⎧√ ⎧ ⎫ √ √1 ⎪ ⎪ − 12 ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪; (b) ⎪ ⎪ ⎪ (a) ⎪ ; ⎪ ⎪ ⎪ ⎪ ⎪ √ ⎪ ⎪ ⎩ √1 ⎭ ⎩ ⎭ − √2 2 ⎫ ⎧ − ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ (c) ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ⎭ −5 −5 H = I − β1 vvT for the given β and v (a) β = 90, v = (−10, 8, −4)T ; (b) β = 70, v = (10, 6, 2)T ; (c) β = 15, v = (−5, −3, 4)T (a) β = 90, v = (0, 10, 4, 8)T ; (b) β = 15, v = (0, 0, −5, −1, 2)T (a) H2 H1 A = R, where Hi = I − vi vT , βi i i = 1, 2, and β1 = 12, β2 = 45 ⎧ ⎧ ⎫ ⎫ −4 ⎪ 0⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 9⎪ ⎪ ⎪ 2⎪ v1 = ⎪ =⎪ ⎪ ⎪ ⎪ ⎩ ⎩ ⎭, v2 ⎪ ⎭, −2 −3 ⎫ ⎧ 19 ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ R=⎪ ⎪, ⎪ ⎪ ⎭ ⎩ −5 −3 ⎪ 0 ⎧ 5⎫ − ⎪ ⎪ ⎪ ⎪ 2⎪ ⎪ ⎪; c = H2 H1 b = ⎪ ⎪ ⎪ −5 ⎪ ⎪ ⎩ ⎭ (b) x = (−4, 1, 0)T ⎫ ⎧ 4⎪ ⎪ ⎪ ⎪ ⎧ ⎫ ⎪ ⎪ ⎪5 5⎪ ⎪ ⎪ ⎪ ⎭ ⎩ −1 ⎪ ⎪ (a) G = ⎪ , x = ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ −3 ⎪ 5 It takes three multiplications, two additions, and one square root to determine H It takes four multiplications/divisions, one addition, and one square root to determine G The calculation of GA requires 4n multiplications 498 Answers to Selected Exercises 10 11 7.6 and 2n additions, while the calculation of HA requires 3n multiplications/divisions and 3n additions (a) n − k + multiplications/divisions, 2n − 2k + additions; (b) n(n − k + 1) multiplications/divisions, n(2n − 2k + 1) additions (a) 4(n − k) multiplications/divisions, 2(n − k) additions; (b) 4n(n − k) multiplications, 2n(n − k) additions (a) rotation; (b) rotation; (c) Givens transformation; (d) Givens transformation ⎧ ⎫ ⎧ ⎫ 1⎪ 0⎪ ⎪ ⎪ ⎩ ⎭ ⎩ ⎭; (a) u1 = ; (b) A2 = 0 (c) λ1 = 2, λ2 = 0; the eigenspace corresponding to λ1 is spanned by u1 ⎧ ⎫ ⎧ ⎫ 3⎪ 0.6 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪, u1 = ⎪ ⎪ 5⎪ 1.0 ⎪ (a) v1 = ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎩ ⎭ ⎭, 0.6 ⎧ ⎧ ⎫ ⎫ ⎪ ⎪ ⎪ 2.2 ⎪ ⎪ 0.52 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v2 = ⎪ ⎪ 4.2 ⎪ ⎪ 1.00 ⎪ ⎪, u2 = ⎪ ⎪, ⎩ ⎩ ⎭ ⎭ 2.2 0.52 ⎧ ⎫ 2.05 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 4.05 ⎪ v3 = ⎪ ⎪ ⎪ ⎩ ⎭; 2.05 (b) λ1 = 4.05; (c) λ1 = 4, δ = 0.0125 (b) A has no dominant eigenvalue ⎫ ⎫ ⎧ ⎧ ⎭, A3 = ⎪ ⎭, ⎩ −1 ⎪ ⎩ 3.4 0.2 ⎪ A2 = ⎪ −1 0.2 0.6 √ √ λ1 = + ≈ 3.414, λ2 = − ≈ 0.586 (b) H = I − vvT , where β = 13 and v = β (− 13 , − 32 , 13 )T ; 7.7 ⎧ ⎫ 3⎪ ⎪ ⎪ ⎪ −4 ⎪ ⎪ ⎪ (c) λ2 = 3, λ3 = 1, HAH = ⎪ ⎪ ⎪ ⎩ ⎭ −1 √ √ √ √ (a) ( 2, 0)T ; (b) (1 − 2, 2, − 2)T ; √ √ √ (c) (1, 0)T ; (d) (1 − 2, 2, − 2)T di bi + ei bn+i xi = , i = 1, , n di2 + e2i ⎫ ⎧1 − 16 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎫ ⎧ ⎪ ⎪ −2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ 12 ⎪ (a) Q = ⎪ , R = ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 12 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩1 ⎭ − ⎧ (b) x = ⎩ ⎫T ⎭ (a) σ1 = + ρ 2, σ2 = ρ; √ (b) λ1 = 2, λ2 = 0, σ1 = 2, σ2 = ⎫ ⎧1 0⎪ ⎪ ⎪ ⎪ 4 ⎪ ⎪ + ⎪ ⎪ 12 A = ⎪ ⎪ ⎭ ⎩1 4 ⎧ ⎫ − 10 ⎪ ⎪ ⎪ ⎪ 10 ⎪ ⎪ ⎪; 13 (a) A+ = ⎪ ⎪ ⎪ ⎩ 2 ⎭ − 10 10 ⎧ ⎫ ⎭; ⎩1⎪ (b) A+ b = ⎪ ⎫ ⎧ ⎧ ⎫ ⎭ ⎭+α⎪ ⎩ −2 ⎪ ⎩1⎪ (c) y y = ⎪ 15 + A1 −A2 F = ρ, A+ −A2 F = 1/ρ As ρ → + 0, A1 − A2 F → and A+ − A2 F → ∞ CHAPTER TEST A False False False True False False True False False 10 False INDEX A Absolute error, 397 Addition of matrices, 29 in Rn , 114 of vectors, 115 Adjacency matrix, 57 Adjoint of a matrix, 101 Aerospace, 188, 294 Analytic hierarchy process, 38, 234, 382, 443 Angle between vectors in 2-space, 202 Angle between vectors, 44, 106, 208 Approximation of functions, 256–259 Astronomy Ceres orbit of Gauss, 226 ATLAST, xiv, 471 Augmented matrix, Automobile leasing, 316 Aviation, 188 B Backslash operator, 474 Back substitution, 5, 407, 408 Basis, 141 change of, 147–157 orthonormal, 249 Bidiagonalization, 458 Binormal vector, 108 Block multiplication, 72–76 C C[a, b], 116 Catastrophic cancellation, 401 Cauchy–Schwarz inequality, 204, 243 Characteristic equation, 291 Characteristic polynomial, 291 Characteristic value(s), 290 Characteristic vector, 290 Chebyshev polynomials, 279 of the second kind, 282 Chemical equations, 20 Cholesky decomposition, 374 Closure properties, 115 Cn , 330 Coded messages, 104–105 Coefficient matrix, Cofactor, 90 Cofactor expansion, 90 Column space, 157, 219 Column vector notation, 28 Column vector(s), 27, 157 Communication networks, 56 Companion matrix, 300 Comparison matrix, 383 Compatible matrix norms, 416 Complete pivoting, 413 Complex eigenvalues, 296, 305–306 matrix, 331 Computer graphics, 185 Condition number, 421–426 formula for, 423 Conic sections, 357–363 Consistency Theorem, 34, 158 Consistent comparison matrix, 384 Consistent linear system, Contraction, 185 Cooley, James W., 262 Coordinate metrology, 232 Coordinate vector, 147, 153 Coordinates, 153 Correlation matrix, 213 Correlations, 211 Covariance, 213 Covariance matrix, 214 Cramer’s rule, 103 Cross product, 105 Cryptography, 104–105 D Dangling Web page, 320 Data fitting, least squares, 229–232 Defective matrix, 315 Definite quadratic form, 364 Deflation, 445 Determinant(s), 87–111 cofactor expansion, 90 definition, 92 and eigenvalues, 291 of elementary matrices, 97 and linear independence, 135 of a product, 99 of a singular matrix, 97 of the transpose, 92 of a triangular matrix, 93 DFT, 261 Diagonal matrix, 66 Diagonalizable matrix, 312 Diagonalizing matrix, 312 Digital imaging, 352 Dilation, 185 Dimension, 143 of row space and column space, 160 Dimension Theorem, 275 Direct sum, 221 Discrete Fourier transform, 259–261 Distance in 2-space, 202 in n-space, 208, 246 in a normed linear space, 245 Dominant eigenvalue, 319 E Economic models, 21–23 Edges of a graph, 56 Eigenspace, 291 Eigenvalue(s), 290 complex, 296 definition, 290 499 500 Index and determinants, 291 numerical computation, 440–451 product of, 297 sensitivity of, 464 of similar matrices, 298 and structures, 293, 388 sum of, 297 of a symmetric positive definite matrix, 365 Eigenvector, 290 Electrical networks, 19 Elementary matrix, 61 determinant of, 97 inverse of, 63 Equivalent systems, 3–5, 61 Euclidean length, 202 Euclidean n-space, 27 F Factor analysis, 214 Fast Fourier Transform, 262–263 Filter bases, 440 Finite dimensional, 143 Floating point number, 396 FLT axis system, 189 Forward substitution, 407, 408 Fourier coefficients, 258 complex, 259 Fourier matrix, 261 Francis, John G F., 447 Free variables, 13 Frobenius norm, 241, 415 Frobenius theorem, 381 Full rank, 165 Fundamental subspaces, 218–219 Fundamental Subspaces Theorem, 219 G Gauss, Carl Friedrich, 225 Gauss–Jordan reduction, 17 Gaussian elimination, 13 algorithm, 405 algorithm with interchanges, 411 complete pivoting, 413 with interchanges, 409–414 without interchanges, 404–409 partial pivoting, 413 Gaussian quadrature, 281 Gerschgorin disks, 467 Gerschgorin’s theorem, 450 Givens transformation, 465 Golub, Gene H., 458 Golub-Reinsch Algorithm, 459 Google PageRank algorithm, 320 Gram–Schmidt process, 266–275 modified version, 273 Graph(s), 56 H Harmonic motion, 308 Hermite polynomials, 279 Hermitian matrix, 332 eigenvalues of, 332 Hessian, 368 Hilbert matrix, 464 Homogeneous coordinates, 187 Homogeneous system, 20 nontrivial solution, 20 Hotelling, H., 354 Householder QR factorization, 453 Householder transformation, 430–435, 465 I Idempotent, 59, 299 Identity matrix, 52 IEEE floating point standard, 400 Ill conditioned, 421 Image space, 175 Inconsistent, Indefinite quadratic form, 364 Infinite dimensional, 143 Information retrieval, 41, 209, 320, 353 Initial value problems, 302, 307 Inner product, 77, 238 complex inner product, 330 for Cn , 331 of functions, 239 of matrices, 239 of polynomials, 239 of vectors in Rn , 238 Inner product space, 238 complex, 330 norm for, 244 Interpolating polynomial, 229 Lagrange, 280 Invariant subspace, 300, 336 Inverse computation of, 65 of an elementary matrix, 63 of a product, 54 Inverse matrix, 53 Inverse power method, 450 Invertible matrix, 53 Involution, 59 Irreducible matrix, 380 Isomorphism between row space and column space, 223 between vector spaces, 119 J Jacobi polynomials, 279 Jordan canonical form, 319 K Kahan, William, 458 Kernel, 175 Kirchhoff’s laws, 19 L Lagrange’s interpolating formula, 280 Laguerre polynomials, 279 Latent semantic indexing, 211 LDLT factorization, 374 LDU factorization, 373 Lead variables, 13 Leading principal submatrix, 370 Index Least squares problem(s), 225–238, 253, 451–462 Ceres orbit of Gauss, 226 fitting circles to data, 232 Least squares problem(s), solution of, 226 by Householder transformations, 453–454 from Gram–Schmidt QR, 271, 452–453 from normal equations, 228, 451 from singular value decomposition, 454–457 Left inverse, 164 Left singular vectors, 345 Legendre polynomials, 278 Legendre, Adrien-Marie, 225 Length of a complex scalar, 330 in inner product spaces, 240 of a vector in Cn , 330 of a vector in R2 , 106, 113, 202 of a vector in Rn , 208 Length of a walk, 57 Leontief input-output models closed model, 23, 381–382 open model, 378–380 Leslie matrix, 51 Leslie population model, 51 Linear combination, 34, 123 Linear differential equations first order systems, 301–306 higher order systems, 306–310 Linear equation, Linear operator, 170 Linear system(s), equivalent, 61 homogeneous, 20 inconsistent, matrix representation, 32 overdetermined, 14 underdetermined, 15 Linear transformation(s), 169–198 contraction, 185 definition, 169 dilation, 185 image space, 175 inverse image, 178 kernel, 175 one-to-one, 178 onto, 178 on R2 , 170 range, 175 reflection, 185 from Rn to Rm , 173 standard matrix representation, 179 Linearly dependent, 132 Linearly independent, 132 in C(n−1) [a, b], 138–140 in Pn , 137–138 Loggerhead sea turtle, 50, 83 Lower triangular, 66 LU factorization, 67, 406 M Machine epsilon, 351, 399, 401 Management Science, 38 Markov chain(s), 45, 149, 316–319, 382 Markov process, 45, 149, 316 MATLAB, 471–482 array operators, 479 built in functions, 476 entering matrices, 472 function files, 477 graphics, 479 help facility, 81, 481 M-files, 476 programming features, 476 relational and logical operators, 478 script files, 476 submatrices, 473 symbolic toolbox, 480 MATLAB path, 477 Matrices addition of, 29 equality of, 29 multiplication of, 35 row equivalent, 64 scalar multiplication, 29 similar, 195 Matrix coefficient matrix, column space of, 157 condition number of, 423 correlation, 213 defective, 315 definition of, determinant of, 92 diagonal, 66 diagonalizable, 312 diagonalizing, 312 elementary, 61 Fourier, 261 Hermitian, 332 identity, 52 inverse of, 53 invertible, 53 irreducible, 380 lower triangular, 66 negative definite, 364 negative semidefinite, 364 nonnegative, 377 nonsingular, 53 normal, 339 null space of, 122 orthogonal, 251 positive, 377 positive definite, 364 positive semidefinite, 364 powers of, 49 projection, 228, 255 rank of, 158 reducible, 380 row space of, 157 singular, 54 sudoku matrix, 427 symmetric, 41 transpose of, 41 triangular, 66 unitary, 333 upper Hessenberg, 446 upper triangular, 66 Matrix algebra, 46–58 algebraic rules, 47 notational rules, 40 Matrix arithmetic, 27–45 501 502 Index Matrix exponential, 323 Matrix factorizations Cholesky decomposition, 374 Gram–Schmidt QR, 269 LDLT , 374 LDU, 373 LU factorization, 67, 406 QR factorization, 434, 437, 453 Schur decomposition, 334 singular value decomposition, 342 Matrix generating functions, 473 Matrix multiplication, 35 definition, 35 Matrix norms, 415–421 1-norm, 379, 418 2-norm, 420 compatible, 416 Frobenius, 241, 415 infinity norm, 418 subordinate, 416 Matrix notation, 27 Matrix representation theorem, 182 Matrix, adjoint of, 101 Maximum local, 368 of a quadratic form, 365 Minimum local, 368 of a quadratic form, 365 Minor, 90 Mixtures, 303 Modified Gram–Schmidt process, 273, 452 Moore–Penrose pseudoinverse, 456 Multipliers, 406 N Negative correlation, 213 Negative definite matrix, 364 quadratic form, 364 Negative semidefinite matrix, 364 quadratic form, 364 Networks communication, 56 electrical, 19 Newtonian mechanics, 106 Nilpotent, 299 Nonnegative matrices, 377–387 Nonnegative matrix, 377 Nonnegative vector, 377 Nonsingular matrix, 53, 64 Norm 1-norm, 244 in Cn , 331 infinity, 244 from an inner product, 240, 244 of a matrix, 416 of a vector, 244 Normal equations, 228, 451 Normal matrices, 338–339 Normal vector, 206 Normed linear space, 244 Nth root of unity, 265 Null space, 122 dimension of, 159 Nullity, 159 Numerical integration, 280 Numerical rank, 350–351 O Ohm’s law, 19 Operation count evaluation of determinant, 98–99, 101 forward and back substitution, 408 Gaussian elimination, 405 QR factorization, 435, 438 Ordered basis, 147 Origin shifts, 449 Orthogonal complement, 218 Orthogonal matrices, 251–253 definition, 251 elementary, 430 Givens reflection, 435, 437 Householder transformation, 430–435 permutation matrices, 252 plane rotation, 435, 437 properties of, 252 Orthogonal polynomials, 275–282 Chebyshev polynomials, 279 definition, 276 Hermite, 279 Jacobi polynomials, 279 Laguerre polynomials, 279 Legendre polynomials, 278 recursion relation, 277 roots of, 281 Orthogonal set(s), 247 Orthogonal subspaces, 217 Orthogonality in n-space, 208 in an inner product space, 240 in R2 or R3 , 204 Orthonormal basis, 249 Orthonormal set(s), 247–266 Outer product, 77 Outer product expansion, 77 from singular value decomposition, 349, 353 Overdetermined, 14 P PageRank algorithm, 320 Parseval’s formula, 250 Partial pivoting, 413 Partitioned matrices, 70–76 Pascal matrix, 392 Pearson, Karl, 354 Penrose conditions, 455 Permutation matrix, 252 Perron’s theorem, 380 Perturbations, 395 Pitch, 188 Pivot, Plane equation of, 206 Plane rotation, 435, 437 Pn , 117 Population migration, 148 Positive correlation, 213 Index Positive definite matrix, 370–377 Cholesky decomposition, 374 definition, 364 determinant of, 370 eigenvalues of, 365 LDLT factorization, 374 leading principal submatrices of, 370 Positive definite quadratic form, 364 Positive matrix, 377 Positive semidefinite matrix, 364 quadratic form, 364 Positive vector, 377 Power method, 442 Principal Axes Theorem, 363 Principal component analysis, 214, 215, 354 Probability vector, 317 Projection onto column space, 227 onto a subspace, 255 Projection matrix, 228, 255 Pseudoinverse, 455 Psychology, 214 Pythagorean law, 208, 240 Q QR algorithm, 448–449 QR factorization, 269, 434, 437, 453 Quadratic equation in n variables, 362 in two variables, 357 Quadratic form in n variables, 362 negative definite, 364 negative semidefinite, 364 positive definite, 364 positive semidefinite, 364 in two variables, 357 R Rm×n , 115 Rn , 27 Range, 175 of a matrix, 219 Rank deficient, 165 Rank of a matrix, 158 Rank-Nullity Theorem, 159 Rayleigh quotient, 341 Real Schur decomposition, 336 Real Schur form, 336 Reciprocal matrix, 383 Reduced row echelon form, 16 Reducible matrix, 380 Reflection, 185 Reflection matrix, 435, 437 Regular Markov process, 319, 382 Relative error, 397 Relative residual, 422 Residual vector, 226 Right inverse, 164 Right singular vectors, 345 Roll, 188 Rotation matrix, 180, 435, 437, 465 Round off error, 397 Row echelon form, 13 Row equivalent, 64 Row operations, 5, Row space, 157 Row vector notation, 28 Row vector(s), 27, 157 S Saddle point, 365, 368 Scalar multiplication for matrices, 29 in Rn , 114 in a vector space, 115 Scalar product, 31, 77, 202 in R2 or R3 , 202–205 Scalar projection, 205, 242 Scalars, 27 Schur decomposition, 334 Schur’s theorem, 334 Sex-linked genes, 321, 390 Signal processing , 259–262 Similarity, 192–198, 298 definition, 195 eigenvalues of similar matrices, 298 503 Singular matrix, 54 Singular value decomposition, 44, 211, 215, 342, 466 compact form, 345 and fundamental subspaces, 345 and least squares, 454 and rank, 345 Singular values, 342 and 2-norm, 420 and condition number, 421 and the Frobenius norm, 347 Skew Hermitian, 338, 341 Skew symmetric, 101, 338 Solution set of linear system, Space shuttle, 294 Span, 123 Spanning set, 125 Spearman, Charles, 214 Spectral Theorem, 335 Square matrix, Stable algorithm, 395 Standard basis, 145–146 for Pn , 146 for R2×2 , 145 for R3 , 141 for Rn , 145 State vectors, 317 Stationary point, 363 Steady-state vector, 289 Stochastic matrix, 149, 317 Stochastic process, 316 Strict triangular form, Subordinate matrix norms, 416 Subspace(s), 119–130 definition, 120 Sudoku, 427 Sudoku matrix, 427 Svd, 342 Sylvester’s equation, 341 Symmetric matrix, 41 T Trace, 198, 247, 297 Traffic flow, 17 Transition matrix, 150, 154 for a Markov process, 317 Translations, 186 504 Index Transpose of a matrix, 41 of a product, 55 Triangle inequality, 244 Triangular factorization, 67–68, 406 Triangular matrix, 66 Trigonometric polynomial, 258 Trivial solution, 20 Tukey, John W., 262 U Uncorrelated, 213 Underdetermined, 15 Uniform norm, 244 Unit lower triangular, 373 Unit round off, 351 Unit triangular, 373 Unit upper triangular, 373 Unit vector, 106 Unitary matrix, 333 Upper Hessenberg matrix, 446 Upper triangular, 66 V Vandermonde matrix, 70, 100 in MATLAB, 100, 463 Vector projection, 205, 242 Vector space axioms of , 115 closure properties, 115 of continuous functions, 116 definition, 115 of m × n matrices, 115 of polynomials, 117 subspace of, 120 Vector(s), 27 Vectors in Rn , 27 Vertices of a graph, 56 Vibrations of a structure, 310 W Walk in a graph, 57 Wavelets, 440 Web searches, 44, 320 Weight function, 239 Weights, 238 Well conditioned, 421 Wronskian, 139 Y Yaw, 188 Z Zero matrix, 30 subspace, 120 vector, 115 ... Neidinger • • • • Linear Algebra Labs with MATLAB: 3rd ed by David Hill and David Zitarelli Visualizing Linear Algebra using Maple, by Sandra Keith A Maple Supplement for Linear Algebra, by John... a list of some of the companion books being offered as bundles with this textbook: • Student Study Guide for Linear Algebra with Applications The manual is available to students as a study tool... algebra education One of the recommendations of the Linear Algebra Curriculum Study Group is that technology should be used in a first course in linear algebra That recommendation has been widely accepted,