Linear Algebra with Applications 9th Edition Solution Manual - Steven J. Leon - Complete Instructor's Guide

INSTRUCTOR’S SOLUTION V



S MANUAL
V




LINEAR A LGEBRA WIT V V



H A PPLICATIONS
V



NINTH EDITION V




Steven J. Leon V V

UniversityVofVMassachusetts,VDartmouth




BostonV ColumbusV IndianapolisV NewVYorkV SanVFrancisco
AmsterdamV CapeVTownV DubaiV LondonV MadridV MilanV MunichV ParisV MontrealV Tor
ontoVDelhiV MexicoVCityV SãoVPauloV SydneyV HongVKongV SeoulV SingaporeV TaipeiV T
okyo

,TheVauthorVandVpublisherVofVthisVbookVhaveVusedVtheirVbestVeffortsVinVpreparingVthisVbook.VTheseVeffortsV
includeVtheVdevelopment, Vresearch,VandVtestingVofVtheVtheoriesVandVprogramsVtoVdetermineVtheirVeffectivenes
s.VTheVauthorVandVpublisherVmakeVnoVwarrantyVofVanyVkind,VexpressedVorVimplied,VwithVregardVtoVtheseVp
rogramsVorVtheVdocumentationVcontainedVinVthisVbook.VTheVauthorVandVpublisherVshallVnotVbeVliableVinVan
yVeventVforVincidentalVorVconsequential VdamagesVinVconnectionVwith,VorVarisingVoutVof,VtheVfurnishing,Vperf
ormance,VorVuseVofVtheseVprograms.

ReproducedVbyVPearsonVfromVelectronicVfilesVsuppliedVbyVtheVauthor.V

CopyrightV©V2015,V2010,V2006VPearsonVEducation,VInc.
PublishingVasVPearson,V75VArlingtonVStreet,VBoston,VMAV02116.

AllVrightsVreserved.VNoVpartVofVthisVpublicationVmayVbeVreproduced,VstoredVinVaVretrievalVsystem,VorVtrans
mitted,VinVanyVformVorVbyVanyVmeans,Velectronic,Vmechanical,Vphotocopying, Vrecording, VorVotherwise,Vwitho
utVtheVpriorVwrittenVpermissionVofVtheVpublisher.VPrintedVinVtheVUnitedVStatesVofVAmerica.

ISBN-13:V978-0-321-98305-
3VISBN-10:V0-321-98305-X

1V2V3V4V5V6VOPMV17V16V15V14

www.pearsonhighered.com

, Contents
Preface v

1 MatricesV andV SystemsV ofV Equations 1
1 SystemsV ofV LinearV Equations 1
2 RowV EchelonV Form 2
3 Matrix V Arithmetic 3
4 Matrix V Algebra 6
5 ElementaryV Matrices 12
6 PartitionedV Matrices 17
MATLAB V Exercises 20
ChapterVTestVA 22
ChapterVTestVB 24

2 Determinants 27
1 TheV DeterminantV ofV aV Matrix 27
2 PropertiesV ofV Determinants 30
3 AdditionalV TopicsV andV Applications 33
MATLAB V Exercises 35
ChapterVTestVA 35
ChapterVTestVB 36

3 VectorV Spaces 38
1 DefinitionV andV Examples 38
2 Subspaces 42
3 LinearV Independence 47
4 BasisV andV Dimension 50
5 ChangeV ofV Basis 52
6 RowV SpaceV andV ColumnV Space 52
MATLAB V Exercises 59
ChapterVTestVA 60
ChapterVTestVB 62

4 LinearV Transformations 66
1 DefinitionV andV Examples 66
2 MatrixV RepresentationsV ofV LinearV Transformations 69
3 Similarity 71
MATLABV Exercise 72

iii




CopyrightV©V2015VPearsonVEducation,
VInc.

, iv Contents



ChapterVTestVA 73
ChapterVTestVB 74

5 Orthogonality 76
1 TheV ScalarV productV inV Rn 76
2 OrthogonalV Subspaces 78
3 LeastV SquaresV Problems 81
4 InnerV ProductV Spaces 85
5 Orthonormal V Sets 90
6 TheV Gram-SchmidtV Process 98
7 OrthogonalV Polynomials 100
MATLAB V Exercises 103
ChapterVTestVA 104
ChapterVTestVB 105

6 Eigenvalues 109
1 EigenvaluesVandVEigenvectors 109
2 SystemsV ofV LinearV DifferentialV Equations 114
3 Diagonalization 115
4 HermitianV Matrices 123
5 SingularV ValueV Decomposition 130
6 QuadraticV Forms 132
7 PositiveV DefiniteV Matrices 135
8 NonnegativeV Matrices 138
MATLAB V Exercises 140
ChapterVTestVA 144
ChapterVTestVB 145

7 NumericalV LinearV Algebra 149
1 Floating-PointV Numbers 149
2 GaussianV Elimination 150
3 PivotingV Strategies 151
4 MatrixV NormsV andV ConditionV Numbers 152
5 OrthogonalVTransformations 162
6 TheV EigenvalueV Problem 164
7 LeastV SquaresV Problems 168
MATLAB V Exercises 171
ChapterVTestVA 172
ChapterVTestVB 173




CopyrightV©V2015VPearsonVEducation,
VInc.