Linear Algebra with Applications

529 Pages·2015·7.77 MB·English

by  Steven J. Leon

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Linear Algebra with Applications
Ninth Edition
Global Edition
Steven J. Leon
University of Massachusetts, Dartmouth
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Authorized adaptation from the United States edition, entitled Linear Algebra with Applications, 9th
edition,ISBN978-0-321-96221-8,byStevenJ.Leon,publishedbyPearsonEducation©2015.
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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
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Contents
Preface
9
1
Matrices and Systems of Equations 17
1.1 SystemsofLinearEquations 17
1.2 RowEchelonForm 27
1.3 MatrixArithmetic 43
1.4 MatrixAlgebra 62
1.5 ElementaryMatrices 76
1.6 PartitionedMatrices 86
MATLABExercises 96
ChapterTestA—TrueorFalse 100
ChapterTestB 101
2
Determinants 103
2.1 TheDeterminant ofaMatrix 103
2.2 PropertiesofDeterminants 110
2.3 AdditionalTopicsandApplications 117
MATLABExercises 125
ChapterTestA—TrueorFalse 127
ChapterTestB 127
3
Vector Spaces 128
3.1 DefinitionandExamples 128
3.2 Subspaces 135
3.3 LinearIndependence 146
3.4 BasisandDimension 157
3.5 ChangeofBasis 163
3.6 RowSpaceandColumnSpace 173
MATLABExercises 181
ChapterTestA—TrueorFalse 182
ChapterTestB 183
5
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6 Contents
4
Linear Transformations 185
4.1 DefinitionandExamples 185
4.2 MatrixRepresentationsofLinearTransformations 194
4.3 Similarity 208
MATLABExercises 214
ChapterTestA—TrueorFalse 215
ChapterTestB 216
5
Orthogonality 217
5.1 TheScalarProductinRn 218
5.2 OrthogonalSubspaces 233
5.3 LeastSquaresProblems 241
5.4 InnerProductSpaces 254
5.5 Orthonormal Sets 263
5.6 TheGram–Schmidt Orthogonalization Process 282
5.7 OrthogonalPolynomials 291
MATLABExercises 299
ChapterTestA—TrueorFalse 301
ChapterTestB 301
6
Eigenvalues 303
6.1 EigenvaluesandEigenvectors 304
6.2 SystemsofLinearDifferential Equations 317
6.3 Diagonalization 328
6.4 HermitianMatrices 346
6.5 TheSingularValueDecomposition 358
6.6 QuadraticForms 372
6.7 PositiveDefiniteMatrices 386
6.8 NonnegativeMatrices 393
MATLABExercises 403
ChapterTestA—TrueorFalse 409
ChapterTestB 409
7
Numerical Linear Algebra 411
7.1 Floating-PointNumbers 412
7.2 GaussianElimination 420
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Contents 7
7.3 PivotingStrategies 425
7.4 MatrixNormsandCondition Numbers 431
7.5 OrthogonalTransformations 445
7.6 TheEigenvalueProblem 456
7.7 LeastSquaresProblems 467
MATLABExercises 479
ChapterTestA—TrueorFalse 484
ChapterTestB 484
8 Iterative Methods Online∗
8.1 BasicIterativeMethods
9 Canonical Forms Online∗
9.1 NilpotentOperators
9.2 TheJordanCanonicalForm
Appendix:MATLAB 487
Bibliography 499
AnswerstoSelectedExercises 502
Index 515
∗
Online:ThesupplementalChapters8and9canbedownloadedfromtheInternet.Seethesectionofthe
Prefaceonsupplementarymaterials.
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Preface
Iampleasedtoseethetextreachitsninthedition.Thecontinuedsupportandenthusi-
asmofthemanyusershasbeenmostgratifying.Linearalgebraismoreexcitingnow
thanatalmostanytimeinthepast.Itsapplicationscontinuetospreadtomoreandmore
fields. Largely due to the computer revolution of the last 75 years, linear algebra has
risentoaroleofprominenceinthemathematicalcurriculumrivalingthatofcalculus.
Modernsoftwarehasalsomadeitpossibletodramaticallyimprovethewaythecourse
istaught.
The first edition of this book was published in 1980. Many significant changes
were made for the second edition (1986), most notably the exercise sets were greatly
expanded and the linear transformations chapter of the book was completely revised.
Eachofthefollowingeditionshasseensignificantmodificationsincludingtheaddition
of comprehensive sets of MATLAB computer exercises, a dramatic increase in the
numberofapplications,andmanyrevisionsinthevarioussectionsofthebook.Ihave
been fortunate to have had outstanding reviewers and their suggestions have led to
manyimportantimprovementsinthebook.Forthenintheditionwehavegivenspecial
attentiontoChapter7asitistheonlychapterthathasnotseenmajorrevisionsinany
of the previous editions. The following is an outline of the most significant revisions
thatweremadefortheninthedition.
What’s New in the Ninth Edition?
1. NewSubsectionAddedtoChapter3
Section 2 of Chapter 3 deals with the topic of subspaces. One important example
ofasubspaceoccurswhenwefindallsolutionstoahomogeneoussystemoflinear
equations.Thistypeofsubspaceisreferredtoasanullspace.Anewsubsectionhas
beenaddedtoshowhowthenullspaceisalsousefulinfindingthesolutionsettoa
nonhomogeneouslinearsystem.Thesubsectioncontainsanewtheoremandanew
figurethatprovidesageometricillustrationofthetheorem.Threerelatedproblems
havebeenaddedtotheexercisesattheendofSection2.
2. NewApplicationsAddedtoChapters1,5,6,and7
In Chapter 1, we introduce an important application to the field of Management
Science.Managementdecisionsofteninvolvemakingchoicesbetweenanumberof
alternatives. We assume that the choices are to be made with a fixed goal in mind
and should be based on a set of evaluation criteria. These decisions often involve
a number of human judgments that may not always be completely consistent. The
analytic hierarchy process is a technique for rating the various alternatives based
on a chart consisting of weighted criteria and ratings that measure how well each
alternativesatisfieseachofthecriteria.
9