Linear Algebra, Third Edition: Algorithms, Applications, and Techniques PDF
Preview Linear Algebra, Third Edition: Algorithms, Applications, and Techniques
Linear Algebra Linear Algebra Algorithms, Applications, and Techniques Third Edition Richard Bronson Professor Emeritus ofMathematics Fairleigh Dickinson University Gabriel B. Costa Professor ofMathematical Sciences United StatesMilitary Academy Associate Professor of Mathematics and ComputerScience Seton Hall University John T. Saccoman Professor ofMathematics and Computer Science Seton Hall University AMSTERDAM (cid:129) BOSTON (cid:129) HEIDELBERG (cid:129) LONDON NEW YORK (cid:129) OXFORD (cid:129) PARIS (cid:129) SAN DIEGO SAN FRANCISCO (cid:129) SINGAPORE (cid:129) SYDNEY (cid:129) TOKYO Academic Press is an imprint of Elsevier AcademicPressinanimprintofElsevier 225,WymanStreet,Waltham,MA02451,USA TheBoulevard,LangfordLane,Kidlington,OxfordOX51GB,UK Radarweg29,POBox211,1000AEAmsterdam,TheNetherlands FirstEdition1995 SecondEdition2007 ThirdEditionCopyright©2014,ElsevierInc.Allrightsreserved. Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans, electronicormechanical,includingphotocopy,recording,oranyinformationstorageand retrievalsystem,withoutpermissioninwritingfromthepublisher. PermissionsmaybesoughtdirectlyfromElsevier’sScience&TechnologyRightsDepartmentin Oxford,UK:phone:(þ44)1865843830,fax:(þ44)1865853333,E-mail: permissions@elsevier.com.Youmayalsocompleteyourrequeston-lineviatheElsevier homepage(http://elsevier.com),byselecting“Support&Contact”then“Copyrightand Permission”andthen“ObtainingPermissions.” LibraryofCongressCataloging-inPublicationData Applicationsubmitted BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary ISBN:978-0-12-391420-0 ForinformationonallAcademicPressPublications visitourwebsiteatstore.elsevier.com PrintedandboundinUSA 13 14 15 16 17 10 9 8 7 6 5 4 3 2 1 Dedication To Evy –R.B. TomyteachingcolleaguesatWestPointandSetonHall,especiallytotheGodfather,Dr. v JohnJ.SaccomanandtotheGenerals,Drs.FrankGiordano,ChrisArney,GaryKrahn and Mike Phillips –G.B.C. To Ryan and MaryErin–J.T.S. Contents • PREFACE Ix ABOUT THE AUTHORS xl CHAPTER 1 Matrices 1 CHAPTER 2 VectorSpaces 93 CHAPTER 3 LinearTransformations 175 CHAPTER 4 Eigenvalues, Eigenvectors, and Differential Equations 237 CHAPTER 5 Applications ofEigenvalues 289 CHAPTER 6 Euclidean Inner Product 323 APPENDIX A Jordan Canonical Forms 379 APPENDIX B Markov Chains 413 APPENDIX C More on SpanningTrees ofGraphs 425 APPENDIX D Technology 433 APPENDIX E Mathematical Induction 435 ANSWERS AND HINTS TO SELECTED PROBLEMS 437 INDEX 515 Preface Astechnologyadvances,sodoesourneedtounderstandandcharacterizeit.This isoneofthetraditionalrolesofmathematics,andnowthatwearemorethana decadeintothe21stcentury,noareaofmathematicshasbeenmoreversatilesuc- ix cessfulinthisendeavorthanthatoflinearalgebra.Theelementsoflinearalgebra are the essential underpinnings of a wide range of modern applications, from mathematical modeling in economics to optimization procedures in airline scheduling and inventory control. Linear algebra furnishes today’s analysts in business,engineering,andthesocialscienceswiththetoolstheyneedtodescribe and define thetheories thatdrivetheirdisciplines.Italso provides mathemati- cianswithcompactconstructsforpresentingcentralideasinprobability,differ- ential equations, and operationsresearch. The third edition of this book presents the fundamental structures of linear algebraanddevelopsthefoundationforusingthosestructures.Manyofthecon- ceptsinlinearalgebraareabstract;indeed,linearalgebraintroducesstudentsto formal deductive analysis. Formulating proofs and logical reasoning are skills that require nurturing, and it has been our aim to providethis. Wehavestreamlinedourapproach,inthisthirdedition,whilestrivingtohave the materialpresented in amorelogical and orderly manner. Regarding math- ematical rigor, in the early sections, the proofs are relatively simple, not more than a few lines in length, and deal with concrete structures, such as matrices. Complexity builds as the book progresses. WehavealsointroducedsomegraphtheoreticalconceptsinthiseditionofLinear Algebra. Matrices associated with graphs have been studied extensively, and we attempttointroducethereadertosomeofthesematricesandtheirapplications. Anumberoflearningaidsareincludedtoassistreaders.Newconceptsarecare- fullyintroducedandtiedtothereader’sexperience.Inthebeginning,thebasic conceptsofmatrixalgebraaremadeconcretebyrelatingthemtoastore’sinven- tory. Linear transformations are tied to more familiar functions, and vector spaces areintroducedinthecontextofcolumn matrices.Illustrationsgivegeo- metrical insight on the number of solutions to simultaneous linear equations, vector arithmetic, determinants, and projections tolist just afew. Asinthepreviousedition,wehavehighlightedmaterialtoemphasizeimportant ideasthroughoutthetext.Computationalmethods—forcalculatingtheinverse of a matrix, performing a Gram-Schmidt orthonormalization process, or the like—are presented as a sequence of operational steps. Theorems are clearly marked, and there is a summary of important terms and concepts at the end
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