Elementary Linear Algebra, Fourth Edition

769 Pages·2010·3.5 MB·English

by  Stephen Andrilli

Elementary Linear
Algebra
Fourth Edition
Stephen Andrilli
Department of Mathematics
and Computer Science
La Salle University
Philadelphia, PA
David Hecker
Department of Mathematics
Saint Joseph’s University

Philadelphia, PA
AMSTERDAM•BOSTON•HEIDELBERG•LONDON
NEWYORK•OXFORD•PARIS•SANDIEGO
SANFRANCISCO•SINGAPORE•SYDNEY•TOKYO
AcademicPressisanimprintofElsevier
AcademicPressisanimprintofElsevier
30CorporateDrive,Suite400,Burlington,MA01803,USA
525BStreet,Suite1900,SanDiego,California92101-4495,USA
84Theobald’sRoad,LondonWC1X8RR,UK
Copyright©2010ElsevierInc.Allrightsreserved.
Nopartofthispublicationmaybereproducedortransmittedinanyformorbyanymeans,electronic
ormechanical,includingphotocopy,recording,oranyinformationstorageandretrievalsystem,
withoutpermissioninwritingfromthepublisher.
PermissionsmaybesoughtdirectlyfromElsevier’sScience&TechnologyRightsDepartmentin
Oxford,UK:phone:[phone],fax:[phone],e-mail:[email protected].
YoumayalsocompleteyourrequestonlineviatheElsevierhomepage(http://elsevier.com),by
selecting“Support&Contact”then“CopyrightandPermission”andthen“ObtainingPermissions.”
LibraryofCongressCataloging-in-PublicationData
Applicationsubmitted.
BritishLibraryCataloguing-in-PublicationData
AcataloguerecordforthisbookisavailablefromtheBritishLibrary.
ISBN:978-0-12-374751-8
ForinformationonallAcademicPresspublications
visitourWebsiteat www.elsevierdirect.com

PrintedinCanada
09 10 11 9 8 7 6 5 4 3 2 1
To our wives, Ene and Lyn, for all their help and encouragement
Thispageintentionallyleftblank
Contents
PrefacefortheInstructor ........................................................... ix
PrefacefortheStudent.............................................................. xix
SymbolTable......................................................................... xxiii
ComputationalandNumericalMethods,Applications............................ xxvii
CHAPTER 1 Vectors and Matrices 1
1.1 FundamentalOperationswithVectors .................... 2
1.2 TheDotProduct............................................. 18
1.3 AnIntroductiontoProof Techniques ..................... 31
1.4 FundamentalOperationswithMatrices................... 48
1.5 MatrixMultiplication ....................................... 59
CHAPTER 2 Systems of Linear Equations 79
2.1 SolvingLinearSystemsUsingGaussianElimination ...... 79
2.2 Gauss-JordanRowReductionandReducedRow
EchelonForm ............................................... 98
2.3 EquivalentSystems,Rank,andRowSpace................. 110
2.4 InversesofMatrices......................................... 125
CHAPTER 3 Determinants and Eigenvalues 143
3.1 IntroductiontoDeterminants.............................. 143
3.2 DeterminantsandRowReduction......................... 155
3.3 FurtherPropertiesoftheDeterminant.................... 165
3.4 EigenvaluesandDiagonalization........................... 178
CHAPTER 4 Finite Dime nsional Vector Spaces 203
4.1 IntroductiontoVectorSpaces.............................. 204
4.2 Subspaces.................................................... 215
4.3 Span.......................................................... 227
4.4 LinearIndependence ....................................... 239
4.5 BasisandDimension........................................ 255
4.6 ConstructingSpecialBases................................. 269
4.7 Coordinatization ............................................ 281
CHAPTER 5 Linear Transformations 305
5.1 IntroductiontoLinearTransformations.................... 306
5.2 TheMatrixofaLinearTransformation..................... 321
v
vi Contents
5.3 TheDimensionTheorem ................................... 338
5.4 One-to-OneandOntoLinearTransformations ............ 350
5.5 Isomorphism ................................................ 356
5.6 DiagonalizationofLinearOperators....................... 371
CHAPTER 6 Orthogonality 397
6.1 OrthogonalBasesandtheGram-SchmidtProcess ........ 397
6.2 OrthogonalComplements.................................. 412
6.3 OrthogonalDiagonalization................................ 428
CHAPTER 7 Complex Vector Spaces and General Inner Products 445
7.1 Complexn-VectorsandMatrices .......................... 446
7.2 ComplexEigenvaluesandComplexEigenvectors........ 454
7.3 ComplexVectorSpaces..................................... 460
7.4 OrthogonalityinCn ........................................ 464
7.5 InnerProductSpaces....................................... 472
CHAPTER 8 Additional Applications 491
8.1 GraphTheory ............................................... 491
8.2 Ohm’sLaw................................................... 501
8.3 Least-SquaresPolynomials.................................. 504
8.4 MarkovChains .............................................. 512
8.5 HillSubstitution:AnIntroductiontoCodingTheory..... 525
8.6 ElementaryMatrices........................................ 530
8.7 RotationofAxesforConicSections ....................... 537
8.8 ComputerGraphics......................................... 544
8.9 DifferentialEquations....................................... 561
8.10 Least-SquaresSolutionsforInconsistentSystems......... 570
8.11 QuadraticForms ............................................ 578

CHAPTER 9 Numerical Methods 587
9.1 NumericalMethodsforSolvingSystems .................. 588
9.2 LDUDecomposition........................................ 600
9.3 ThePowerMethodforFindingEigenvalues............... 608
9.4 QRFactorization............................................ 615
9.5 SingularValueDecomposition ............................. 623
Appendix A Miscellaneous Proofs 645
ProofofTheorem1.14,Part(1)................................... 645
ProofofTheorem2.4.............................................. 646
ProofofTheorem2.9.............................................. 647
Contents vii
ProofofTheorem3.3,Part(3),Case2............................ 648
ProofofTheorem5.29............................................. 649
ProofofTheorem6.18............................................. 650
Appendix B Functions 653
Functions:Domain,Codomain,andRange ....................... 653
One-to-OneandOntoFunctions.................................. 654
CompositionandInversesofFunctions.......................... 655
Appendix C Complex Numbers 661
Appendix D Answers to Selected Exercises 665
Index 725
Thispageintentionallyleftblank
Preface for the Instructor
Thistextbookisintendedforasophomore-orjunior-levelintroductorycourseinlinear
algebra.Weassumethestudentshavehadatleastonecourseincalculus.
PHILOSOPHY AND FEATURES OF THE TEXT
Clarity of Presentation: We have striven for clarity and used straightforward lan-
guagethroughoutthebook,occasionallysacrificingbrevityforclearandconvincing
explanation. We hope you will encourage students to read the text deeply and
thoroughly.
Helpful Transition from Computation to Theory: In writing this text,our main intention
was to address the fact that students invariably ran into trouble as the largely com-
putational first half of most linear algebra courses gave way to a more theoretical
secondhalf.Inparticular,manystudentsencountereddifficultieswhenabstractvector
spacetopicswereintroduced.Accordingly,wehavetakengreatcaretohelpstudents
master these important concepts.We consider the material in Sections 4.1 through
5.6 (vector spaces and subspaces,span,linear independence,basis and dimension,
coordinatization,lineartransformations,kernelandrange,one-to-oneandontolinear
transformations,isomorphism,diagonalizationoflinearoperators)tobethe“heart”of
thislinearalgebratext.
Emphasis on the Reading and Writing of Proofs: One reason that students have trouble
with the more abstract material in linear algebra is that most textbooks contain few,
ifany,guidelinesaboutreadingandwritingsimplemathematicalproofs.Thisbookis
intendedtoremedythatsituation.Consequently,wehavestudentsworkingonproofs
as quickly as possible. After a discussion of the basic properties of vectors, there
is a special section (Section 1.3) on general proof techniques,with concrete exam-
plesusingthematerialonvec torsfromSections1.1and1.2.Theearlyplacementof
Section1.3helpstobuildthestudents’confidenceandgivesthemastrongfoundation
inthereadingandwritingofproofs.
We have written the proofs of theorems in the text in a careful manner to give
studentsmodelsforwritingtheirownproofs.Weavoided“clever”or“sneaky”proofs,
in which the last line suddenly produces“a rabbit out of a hat,”because such proofs
invariablyfrustratestudents.Theyaregivennoinsightintothestrategyoftheproof
or how the deductive process was used. In fact,such proofs tend to reinforce the
students’mistakenbeliefthattheywillneverbecomecompetentintheartofwriting
proofs.Inthistext,proofslongerthanoneparagraphareoftenwrittenina“top-down”
manner,a concept borrowed from structured programming.A complex theorem is
brokendownintoasecondaryseriesofresults,whichtogetheraresufficienttoprove
theoriginaltheorem.Inthisway,thestudenthasaclearoutlineofthelogicalargument
andcanmoreeasilyreproducetheproofifcalledontodoso. ix