A Modern Introduction to Linear Algebra © 2010 by Taylor & Francis Group, LLC © 2010 by Taylor & Francis Group, LLC A Modern Introduction to Linear Algebra HENRY RICARDO © 2010 by Taylor & Francis Group, LLC MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2010 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20110720 International Standard Book Number-13: 978-1-4398-9461-3 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the valid- ity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or uti- lized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopy- ing, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com © 2010 by Taylor & Francis Group, LLC For my wife, Catherine—‘‘Age cannot wither her, h nor custom stale her infinite variety.’’— i and for Henry and Marta Ricardo Cathy and Mike Corcoran Tom(cid:1)as and Nicholas Ricardo Christopher Corcoran Christine and Greg Gritmon © 2010 by Taylor & Francis Group, LLC © 2010 by Taylor & Francis Group, LLC Contents Anasterisk * marks optionalsections, those discussions notneededfor later work. Author. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii 1 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1 1.1 Vectors inRn 1 1.1.1 Euclidean n-Space 3 1.1.2 Vector Addition=Subtraction 4 1.1.3 Scalar Multiplication 6 1.1.4 GeometricVectors in R2 and R3 7 1.1.5 AlgebraicProperties 8 1.2 The Inner Productand Norm 12 1.2.1 The Angle between Vectors 16 1.3 SpanningSets 27 1.4 Linear Independence 36 1.5 Bases 45 1.5.1 Coordinates=Change of Basis 53 1.6 Subspaces 58 1.7 Summary 64 2 Systems of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67 2.1 The Geometry ofSystems of Equationsin R2 and R3 68 2.2 Matrices and EchelonForm 77 2.2.1 The Matrix of Coefficients 77 2.2.2 ElementaryRow Operations 79 2.3 Gaussian Elimination 91 *2.4 Computational Considerations—Pivoting 97 2.5 Gauss–Jordan Elimination and Reduced Row EchelonForm 104 *2.6 Ill-Conditioned Systems of Linear Equations 116 2.7 Rank and Nullity ofa Matrix 123 2.7.1 Row Spaces and Column Spaces 123 2.7.2 The Null Space 132 2.8 Systems ofm Linear Equations in n Unknowns 137 2.8.1 The Solution ofLinear Systems 138 2.9 Summary 154 vii © 2010 by Taylor & Francis Group, LLC viii Contents 3 Matrix Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .157 3.1 Addition and Subtraction of Matrices 157 3.1.1 Scalar Multiplication 160 3.1.2 Transpose of aMatrix 161 3.2 Matrix–Vector Multiplication 165 3.2.1 Matrix–Vector Multiplication as a Transformation 172 3.3 The Productof Two Matrices 182 3.3.1 The Column View of Multiplication 185 3.3.2 The Row View of Multiplication 188 3.3.3 Positive Powers ofa SquareMatrix 193 3.4 Partitioned Matrices 203 3.5 Inverses ofMatrices 210 3.5.1 Negative Powers of aSquare Matrix 217 3.6 Elementary Matrices 220 3.7 The LUFactorization 234 3.8 Summary 249 4 Eigenvalues, Eigenvectors, and Diagonalization. . . . . . . . . . . . . .253 4.1 Determinants 254 4.1.1 Cramer’s Rule 268 *4.2 Determinants and Geometry 277 4.3 The Manual Calculation of Determinants 284 4.4 Eigenvalues and Eigenvectors 293 4.5 Similar Matrices and Diagonalization 311 4.6 Algebraicand GeometricMultiplicities of Eigenvalues 321 *4.7 The Diagonalization ofRealSymmetric Matrices 326 4.8 The Cayley–Hamilton Theorem (aFirst Look)=the Minimal Polynomial 333 4.9 Summary 344 5 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .347 5.1 Vector Spaces 347 5.2 Subspaces 353 5.2.1 The Sum and Intersection ofSubspaces 357 5.3 Linear Independence and theSpan 365 5.4 Bases and Dimension 373 5.5 Summary 386 6 Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .389 6.1 Linear Transformations 389 6.2 The Rangeand NullSpace ofa Linear Transformation 399 6.3 The Algebra of Linear Transformations 408 6.4 Matrix Representation of aLinearTransformation 414 6.5 InvertibleLinear Transformations 425 6.6 Isomorphisms 434 6.7 Similarity 441 6.8 SimilarityInvariants ofOperators 453 6.9 Summary 457 © 2010 by Taylor & Francis Group, LLC Contents ix 7 Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .461 7.1 Complex Vector Spaces 461 7.2 Inner Products 468 7.3 Orthogonality and OrthonormalBases 477 7.4 The Gram–SchmidtProcess 483 7.5 Unitary Matrices and OrthogonalMatrices 493 7.6 Schur Factorizationand theCayley–HamiltonTheorem 505 7.7 The QRFactorization and Applications 512 7.8 Orthogonal Complements 520 7.9 Projections 530 7.10 Summary 535 8 Hermitian Matrices and Quadratic Forms . . . . . . . . . . . . . . . . . .539 8.1 Linear Functionalsand theAdjoint of an Operator 539 8.2 HermitianMatrices 547 8.3 Normal Matrices 554 *8.4 Quadratic Forms 560 *8.5 Singular Value Decomposition 569 8.6 The Polar Decomposition 579 8.7 Summary 585 Appendix A: Basics of Set Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . .587 Appendix B: Summation and Product Notation . . . . . . . . . . . . . . . . . . .593 Appendix C: Mathematical Induction. . . . . . . . . . . . . . . . . . . . . . . . . .597 Appendix D: Complex Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .601 Answers=Hints to Odd-Numbered Problems. . . . . . . . . . . . . . . . . . . . . . .605 Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .645 © 2010 by Taylor & Francis Group, LLC © 2010 by Taylor & Francis Group, LLC