Linear Algebra: Ideas and Applications

513 Pages·2015·4.32 MB·English

by  Richard C. Penney

Linear Algebra

Ideas and Applications

FOURTH EDITION

RICHARD C. PENNEY

LINEAR ALGEBRA

LINEAR ALGEBRA

Ideas and Applications

Fourth Edition

RICHARD C. PENNEY

PurdueUniversity
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LibraryofCongressCataloging-in-PublicationData:

Penney,RichardC.

Linearalgebra:ideasandapplications/RichardPenney.–Fourthedition.

pagescm
Includesindex.
ISBN978-1-118-90958-4(cloth)
1.Algebras,Linear–Textbooks. I.Title.
QA184.2.P462015
512′.5–dc23

2015016650

PrintedintheUnitedStatesofAmerica.
10 9 8 7 6 5 4 3 2 1
CONTENTS

PREFACE XI
FEATURESOFTHETEXT XIII

ACKNOWLEDGMENTS XVII
ABOUTTHECOMPANIONWEBSITE XIX

1 SYSTEMSOFLINEAREQUATIONS 1

1.1 TheVectorSpaceofm×nMatrices / 1
TheSpaceRn / 4
LinearCombinationsandLinearDependence / 6

WhatisaVectorSpace? / 11

Exercises / 17

1.1.1 ComputerProjects / 22

1.1.2 ApplicationstoGraphTheoryI / 25
Exercises / 27

1.2 Systems / 28
Rank:TheMaximumNumberofLinearlyIndependent
Equations / 35

Exercises / 38
1.2.1 ComputerProjects / 41
1.2.2 ApplicationstoCircuitTheory / 41

Exercises / 46
v
vi CONTENTS
1.3 GaussianElimination / 47
SpanninginPolynomialSpaces / 58
ComputationalIssues:Pivoting / 61

Exercises / 63
ComputationalIssues:CountingFlops / 68
1.3.1 ComputerProjects / 69
1.3.2 ApplicationstoTrafficFlow / 72
1.4 ColumnSpaceandNullspace / 74

Subspaces / 77

Exercises / 86
1.4.1 ComputerProjects / 94
ChapterSummary / 95

2 LINEARINDEPENDENCEANDDIMENSION 97
2.1 TheTestforLinearIndependence / 97
BasesfortheColumnSpace / 104
TestingFunctionsforIndependence / 106
Exercises / 108
2.1.1 ComputerProjects / 113

2.2 Dimension / 114
Exercises / 123

2.2.1 ComputerProjects / 127

2.2.2 ApplicationstoDifferentialEquations / 128

Exercises / 131

2.3 RowSpaceandtherank-nullitytheorem / 132
BasesfortheRowSpace / 134
ComputationalIssues:ComputingRank / 142

Exercises / 143

2.3.1 ComputerProjects / 146

ChapterSummary / 147
3 LINEARTRANSFORMATIONS 149
3.1 TheLinearityProperties / 149

Exercises / 157

3.1.1 ComputerProjects / 162
3.2 MatrixMultiplication(Composition) / 164
PartitionedMatrices / 171

ComputationalIssues:ParallelComputing / 172
CONTENTS vii
Exercises / 173
3.2.1 ComputerProjects / 178
3.2.2 ApplicationstoGraphTheoryII / 180

Exercises / 181
3.3 Inverses / 182
ComputationalIssues:ReductionversusInverses / 188
Exercises / 190
3.3.1 ComputerProjects / 195

3.3.2 ApplicationstoEconomics / 197

Exercises / 202
3.4 TheLUFactorization / 203
Exercises / 212

3.4.1 ComputerProjects / 214
3.5 TheMatrixofaLinearTransformation / 215
Coordinates / 215
Isomorphism / 228
InvertibleLinearTransformations / 229
Exercises / 230
3.5.1 ComputerProjects / 235

ChapterSummary / 236
4 DETERMINANTS 238

4.1 DefinitionoftheDeterminant / 238

4.1.1 TheRestoftheProofs / 246

Exercises / 249
4.1.2 ComputerProjects / 251
4.2 ReductionandDeterminants / 252

UniquenessoftheDeterminant / 256

Exercises / 258

4.2.1 Volume / 261
Exercises / 263
4.3 AFormulaforInverses / 264
Exercises / 268

ChapterSummary / 269

5 EIGENVECTORSANDEIGENVALUES 271

5.1 Eigenvectors / 271

Exercises / 279
viii CONTENTS
5.1.1 ComputerProjects / 282
5.1.2 ApplicationtoMarkovProcesses / 283
Exercises / 285

5.2 Diagonalization / 287
PowersofMatrices / 288
Exercises / 290
5.2.1 ComputerProjects / 292
5.2.2 ApplicationtoSystemsofDifferentialEquations / 293

Exercises / 295

5.3 ComplexEigenvectors / 296
ComplexVectorSpaces / 303
Exercises / 304

5.3.1 ComputerProjects / 305
ChapterSummary / 306

6 ORTHOGONALITY 308

6.1 TheScalarProductinRN / 308
Orthogonal/OrthonormalBasesandCoordinates / 312
Exercises / 316

6.2 Projections:TheGram-SchmidtProcess / 318
TheQRDecomposition / 325

UniquenessoftheQRFactorization / 327

Exercises / 328

6.2.1 ComputerProjects / 331

6.3 FourierSeries:ScalarProductSpaces / 333
Exercises / 341
6.3.1 ApplicationtoDataCompression:Wavelets / 344

Exercises / 352

6.3.2 ComputerProjects / 353

6.4 OrthogonalMatrices / 355
HouseholderMatrices / 361
Exercises / 364
DiscreteWaveletTransform / 367

6.4.1 ComputerProjects / 369

6.5 LeastSquares / 370
Exercises / 377
6.5.1 ComputerProjects / 380