Linear Algebra, Third Edition: Algorithms, Applications, and Techniques

532 Pages·2013·7.01 MB·English

by  Richard Bronson

Linear Algebra

Linear Algebra
Algorithms, Applications,
and Techniques
Third Edition
Richard Bronson
Professor Emeritus ofMathematics
Fairleigh Dickinson University
Gabriel B. Costa
Professor ofMathematical Sciences
United StatesMilitary Academy
Associate Professor of Mathematics and ComputerScience
Seton Hall University
John T. Saccoman
Professor ofMathematics and Computer Science
Seton Hall University
AMSTERDAM (cid:129) BOSTON (cid:129) HEIDELBERG (cid:129) LONDON
NEW YORK (cid:129) OXFORD (cid:129) PARIS (cid:129) SAN DIEGO
SAN FRANCISCO (cid:129) SINGAPORE (cid:129) SYDNEY (cid:129) TOKYO
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Dedication
To Evy –R.B.
TomyteachingcolleaguesatWestPointandSetonHall,especiallytotheGodfather,Dr. v
JohnJ.SaccomanandtotheGenerals,Drs.FrankGiordano,ChrisArney,GaryKrahn
and Mike Phillips –G.B.C.
To Ryan and MaryErin–J.T.S.

Contents
•
PREFACE Ix
ABOUT THE AUTHORS xl
CHAPTER 1 Matrices 1
CHAPTER 2 VectorSpaces 93
CHAPTER 3 LinearTransformations 175
CHAPTER 4 Eigenvalues, Eigenvectors, and Differential Equations 237
CHAPTER 5 Applications ofEigenvalues 289
CHAPTER 6 Euclidean Inner Product 323
APPENDIX A Jordan Canonical Forms 379
APPENDIX B Markov Chains 413
APPENDIX C More on SpanningTrees ofGraphs 425
APPENDIX D Technology 433
APPENDIX E Mathematical Induction 435
ANSWERS AND HINTS TO SELECTED PROBLEMS 437
INDEX 515

Preface
Astechnologyadvances,sodoesourneedtounderstandandcharacterizeit.This
isoneofthetraditionalrolesofmathematics,andnowthatwearemorethana
decadeintothe21stcentury,noareaofmathematicshasbeenmoreversatilesuc- ix
cessfulinthisendeavorthanthatoflinearalgebra.Theelementsoflinearalgebra
are the essential underpinnings of a wide range of modern applications, from
mathematical modeling in economics to optimization procedures in airline
scheduling and inventory control. Linear algebra furnishes today’s analysts in
business,engineering,andthesocialscienceswiththetoolstheyneedtodescribe
and define thetheories thatdrivetheirdisciplines.Italso provides mathemati-
cianswithcompactconstructsforpresentingcentralideasinprobability,differ-
ential equations, and operationsresearch.
The third edition of this book presents the fundamental structures of linear
algebraanddevelopsthefoundationforusingthosestructures.Manyofthecon-
ceptsinlinearalgebraareabstract;indeed,linearalgebraintroducesstudentsto
formal deductive analysis. Formulating proofs and logical reasoning are skills
that require nurturing, and it has been our aim to providethis.
Wehavestreamlinedourapproach,inthisthirdedition,whilestrivingtohave
the materialpresented in amorelogical and orderly manner. Regarding math-
ematical rigor, in the early sections, the proofs are relatively simple, not more
than a few lines in length, and deal with concrete structures, such as matrices.
Complexity builds as the book progresses.
WehavealsointroducedsomegraphtheoreticalconceptsinthiseditionofLinear
Algebra. Matrices associated with graphs have been studied extensively, and we
attempttointroducethereadertosomeofthesematricesandtheirapplications.
Anumberoflearningaidsareincludedtoassistreaders.Newconceptsarecare-
fullyintroducedandtiedtothereader’sexperience.Inthebeginning,thebasic
conceptsofmatrixalgebraaremadeconcretebyrelatingthemtoastore’sinven-
tory. Linear transformations are tied to more familiar functions, and vector
spaces areintroducedinthecontextofcolumn matrices.Illustrationsgivegeo-
metrical insight on the number of solutions to simultaneous linear equations,
vector arithmetic, determinants, and projections tolist just afew.
Asinthepreviousedition,wehavehighlightedmaterialtoemphasizeimportant
ideasthroughoutthetext.Computationalmethods—forcalculatingtheinverse
of a matrix, performing a Gram-Schmidt orthonormalization process, or the
like—are presented as a sequence of operational steps. Theorems are clearly
marked, and there is a summary of important terms and concepts at the end