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505 Pages·2009·3.49 MB·English

Linear Algebra with Applications
Eighth Edition
Steven J. Leon
University of Massachusetts, Dartmouth
Boston Columbus Indianapolis NewYork SanFrancisco UpperSaddleRiver
Amsterdam CapeTown Dubai London Madrid Milan Munich Paris Montreal Toronto
Delhi MexicoCity SaoPaulo Sydney HongKong Seoul Singapore Taipei Tokyo
LibraryofCongressCataloging-in-PublicationData
Leon,StevenJ.
Linearalgebrawithapplication/StevenJ.Leon.--8thed.
p.cm.
Includesbibliographicalreferencesandindex.
ISBN978-0-13-600929-0
1.Algebras,Linear--Textbooks.I.Title.
QA184.2.L462010
512.5--dc22
2009023730
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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
ix
1
Matrices and Systems of Equations 1
1.1 Systemsof LinearEquations 1
1.2 RowEchelonForm 11
1.3 MatrixArithmetic 27
1.4 MatrixAlgebra 44
1.5 ElementaryMatrices 58
1.6 PartitionedMatrices 68
(cid:2)
MATLABR Exercises 77
Chapter TestA 81
Chapter TestB 82
2
Determinants 84
2.1 TheDeterminantofa Matrix 84
2.2 PropertiesofDeterminants 91
2.3† AdditionalTopicsandApplications 98
MATLABExercises 106
Chapter TestA 108
Chapter TestB 108
3
Vector Spaces 110
3.1 Definitionand Examples 110
3.2 Subspaces 117
3.3 LinearIndependence 127
3.4 BasisandDimension 138
3.5 ChangeofBasis 144
3.6 RowSpaceandColumnSpace 154
MATLABExercises 162
Chapter TestA 164
Chapter TestB 164
v
vi Contents
4
Linear Transformations 166
4.1 DefinitionandExamples 166
4.2 MatrixRepresentationsof Linear Transformations 175
4.3 Similarity 189
MATLABExercises 195
ChapterTestA 196
ChapterTestB 197
5
Orthogonality 198
5.1 TheScalar ProductinRn 199
5.2 OrthogonalSubspaces 214
5.3 LeastSquaresProblems 222
5.4 InnerProductSpaces 232
5.5 OrthonormalSets 241
5.6 TheGram–SchmidtOrthogonalizationProcess 259
5.7† OrthogonalPolynomials 269
MATLABExercises 277
ChapterTestA 279
ChapterTestB 280
6
Eigenvalues 282
6.1 EigenvaluesandEigenvectors 283
6.2 SystemsofLinearDifferentialEquations 296
6.3 Diagonalization 307
6.4 HermitianMatrices 324
6.5 TheSingularValue Decomposition 337
6.6 QuadraticForms 351
6.7 Positive DefiniteMatrices 364
6.8† NonnegativeMatrices 372
MATLABExercises 378
ChapterTestA 384
ChapterTestB 384
7
Numerical Linear Algebra 386
7.1 Floating-PointNumbers 387
7.2 GaussianElimination 391
Contents vii
7.3 PivotingStrategies 398
7.4 MatrixNormsandConditionNumbers 403
7.5 OrthogonalTransformations 417
7.6 TheEigenvalue Problem 428
7.7 LeastSquaresProblems 437
MATLABExercises 448
Chapter TestA 454
Chapter TestB 454
8
Iterative Methods Web∗
9
Canonical Forms Web∗
Appendix: MATLAB 456
Bibliography 468
AnswerstoSelectedExercises 471
Index 485
†Optionalsections.Thesesectionsarenotprerequisitesforanyothersectionsofthebook.
∗
Web:ThesupplementalChapters8and9canbedownloadedfromtheinternet.Seethesectionofthe
PrefaceonsupplementalWebmaterials.
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Preface
I am pleased to see the text reach its eighth edition. The continued support and en-
thusiasmofthemanyusershasbeenmostgratifying. Linearalgebraismoreexciting
now than at almost any time in the past. Its applications continue to spread to more
andmorefields. Largelyduetothecomputerrevolutionofthelasthalfcentury,linear
algebrahasrisentoaroleofprominenceinthemathematicalcurriculumrivalingthat
of calculus. Modern software has also made it possible to dramatically improve the
way the course is taught. I teach linear algebra every semester and continue to seek
newwaystooptimizestudentunderstanding. Forthisedition,everychapterhasbeen
carefullyscrutinizedandenhanced. Additionally,manyoftherevisionsinthisedition
are due to the helpful suggestions received from users and reviewers. Consequently,
thisnewedition,whileretainingtheessenceofpreviouseditions,incorporatesavariety
ofsubstantiveimprovements.
What's New in the Eighth Edition?
1. NewSectiononMatrixArithmetic
Oneofthelongersectionsinthepreviouseditionwasthesectiononmatrixalgebra
inChapter1. Thematerialinthatsectionhasbeenexpandedfurtherforthecurrent
edition. Rather than include an overly long revised section, we have divided the
materialintosectionstitledMatrixArithmeticandMatrixAlgebra.
2. NewExercises
After seven editions it was quite a challenge to come up with additional original
exercises. However,theeightheditionhasmorethan130newexercises.
3. NewSubsectionsandApplications
A new subsection on cross products has been included in Section 3 of Chapter 2.
AnewapplicationtoNewtonianMechanicshasalsobeenaddedtothatsection. In
Section 4 of Chapter 6 (Hermitian Matrices), a new subsection on the Real Schur
Decompositionhasbeenadded.
4. NewandImprovedNotation
Thestandardnotationforthe jthcolumnvectorofamatrix Aisa ,however,there
j
seems to be no universally accepted notation for row vectors. In the MATLAB
package,theithrowof Aisdenotedby A(i,:). Inpreviouseditionsofthisbookwe
usedasimilarnotationa(i,:);however,thisnotationseemssomewhatartificial. For
thiseditionweusethesamenotationasacolumnvectorexceptweputahorizontal
arrowabovethelettertoindicatethatthevectorisarowvector(ahorizontalarray)
ratherthanacolumnvector(averticalarray). Thustheithrowof Aisnowdenoted
bya(cid:4) .
i
We have also introduced improved notation for the standard Euclidean vector
spacesandtheircomplexcounterparts. WenowusethesymbolsRn andCn inplace
ofthe Rn andCn notationusedinearliereditions.
ix