Elements of Linear Algebra and Matrix Theory

384 Pages·1968·26.473 MB·English

by  John T. Moore

ELEMENTS
OF
LINEAR ALGEBRA
AND MATRIX
THEORY
International Series in
Pure and Applied Mathematics
William Ted Martin and E. H. Spanier, ConsultingEditors
AHLFORS - ComplexAnalysis
BELLMAN - StabilityTheoryof Differential Equations
BUCK -Advanced Calculus
BUSACKER ANDSAATY - Finite Graphsand Networks
CHENEY - Introduction toApproximation Theory
CODDINGTON AND LEVINSON -TheoryofOrdinary Differential
Equations
COHN ~ Conformal Mappingon Riemann Surfaces
DENNEMEYER - Introduction to Partial Differential Equations
and BoundaryValue Problems
DETTMAN - Mathematical Methodsin Physicsand Engineering
EPSTEIN - Partial Differential Equations
GOLOMB ANDSHANKS - ElementsofOrdinary Differential
Equations
GRAVES ' TheTheoryof Functionsof Real Variables
GREENSPAN - Introduction to Partial Differential Equations
GRIFFIN - ElementaryTheoryof Numbers
HAMMING - Numerical MethodsforScientistsand Engineers
HILDEBRAND ~ Introduction to Numerical Analysis
HOUSEHOLDER - Principlesof Numerical Analysis
LASS . Elements of Pureand Applied Mathematics
LASS - Vectorand TensorAnalysis ,
LEPAGE - ComplexVariablesandthe LaplaceTransformfor
Engineers
McCARTY -Topology: An Introduction with Applicationsto
Topological Groups
MOORE . Elementsof LinearAlgebra and MatrixTheory
MOURSUNDAND DURIS . ElementaryTheoryand Application
. of Numerical Analysis
NEF ~ Linear Algebra t"
NEHARI - Conformal Mapping
NEWELL - VectorAnalysis
RALSTON - A First Coursein Numerical Analysis
RITGERAND ROSE - Differential EquationswithApplications
ROSSER - Logicfor Mathematicians
RUDIN - Principlesof Mathematical Analysis
SAATYAND BRAM - Nonlinear Mathematics
SIMMONS - Introduction toTopologyand Modern Analysis
SNEDDON - Elementsof Partial Differential Equations
SNEDDON . FourierTransforms
STOLL - LinearAlgebra and MatrixTheory
STRUBLE- Nonlinear Differential Equations
WEINSTOCK - CalculusofVariations
WEISS - Algebraic NumberTheory
ZEMANIAN - Distribution Theoryand Transform Analysis
ELEMENTS
OF
LINEAR ALGEBRA
AND MATRIX
THEORY
JOHN [MOORE
The University ofFlorida
The University of ‘Western Ontario
McGRAW-HILL BOOK COMPANY
New York
St. Louis
San Francisco
Toronto
London
Sydney
ELEMENTS
OF
LINEAR ALGEBRA
AND MATRIX
THEORY
Copyright ©1968 by McGraw-Hill, Inc. All Rights
Reserved. Printed in the United StatesofAmerica.
No partofthis publication may be reproduced,-
stored in a retrieval system, ortransmitted, in any
form or byanymeans, electronic, mechanical,
photocopying, recording, orotherwise, withoutthe
priorwritten permission ofthe publisher.
LibraryofCongressCatalogCardNumber68-11933
42885
1234567890 MAMM 7543210698
D. E. S.
F. A. S.

PREFACE
This book is an outgrowth of a course in linear algebra which I first gave to a
class of undergraduates several years ago. In writing it, l have been moti-
vated by the desire to produce a text with the following features: It would be
ofa size such thata major portion ofthecontentscould becovered in a single
academic term; it would have an abundance of problems, with degree of
difficulty varying from the routine and near-trivial to the challenging; it would
use a notation and terminology as near-standard as possible; and it would
present a level of mathematics suitable for students whose major interest is
in either pure mathematics or any of the pure or applied sciences. In my
desire to limit the size of the book, it is inevitable that I am exposing an
“Achilles’ heel” by my choice of material for inclusion. However, it is my
belief that in a first course one should avoid being encyclopedic but, rather,
help to give some sense of direction to thestudent; and so I take mystand on
what l have done—butwith apologiestothosewhoseopinions maydifferfrom
mine.
Before beginning any writing of my own, I consulted the bulletins of the
Committee on the Undergraduate Program in Mathematics (CUPM) and
decided to make the course recommendation contained in one of them the
nucleus of this book. However, i have not followed these recommendations
slavishly but have included additional material and been guided by my own
vii
viii PREFACE
preference with regard to arrangement. The book has been structured, for
the most part, asa continuous study, in which each section makes a contribu-
tion to the next. At the same time, there are a few sections that are not
essential to the main developmentofthe text; in each such case this fact has
been made clear by an introductory remark. In order to emphasize the key
purpose ofa section, I have limited the numberof results included in any one
of them to a bare minimum.
As was stated above, I have tried to use standard symbolism and termi-
nology, and so I have madefree useofthefamiliarsymbolsofsettheory. To
be precise, the followingsymbols occur without explanation: E for set mem-
bership; U forset-theoretic union; (N forset-theoretic intersection; C and D
for the two kinds of set inclusion; X for the cartesian product of sets. In
addition, it has been convenientto referto the cardinality or cardinalnumber
of a set in an intuitive sortof way, and I have used [a,b] to denote the interval
of real numbers between aand b, inclusive. Although the book isconsidered
to be self-contained, itisassumed thatthe readerisfamiliarwith thefollowing
basic notions: the division and gcd processes as they arise in the theory of
polynomials; the partitioningof a set effected by an equivalence relation, each
equivalence class being representable by a single element ofthe set; and the
two methods of proof bymathematicalinduction. Ifthe readeris notfamiliar
with these several concepts, he is urged to acquaint himself with the
unfamiliar ones by referring to any book on abstract algebra.
The problem sets have been arranged into three groupings, physically
separated from one another by spaces. The problems in the first groupings
are basic but quite simple, and some instructors may find them unnecessary
fortheirclasses. However, although this may betruein some instances, itis
myjudgmentthatastudentisoftenlostata pointwherehefailstocomprehend
something simple—and almost trivial—but also very important. It is the
purposeofthe early problemsto bringto lightand, hopefully, to eliminateany
such difficulties. The second groupings of problems are complementary to
the material in the text, and all of them should be worked out in detail. In
this way, students may feel more personally involved with the proofs of the
theorems than if these proofs were all presented in final polished form. In
thefinal groupings, l have included other problems of a more general nature,
relevanttothe particularsection understudy. Otherwise, the problems have
not been arranged according to difficulty. Inasmuch as the first groupings
involve problems ofsuch a basic nature, answers or hints have been provided
for most of them at the back of the book. The student is thus given an
opportunity to work these problems—if he feels the need of doing do—quite
independently of whether this is suggested by his instructor and so develop
an early confidence in his ability to grasp the concepts being presented.
PREFACE ix
There are many people to whom I am indebted in the writing of this book.
l was first given much encouragement by a number of reviewers of an early
portion of the manuscript, all of whom were anonymous except for Prof.
George Simmons of Colorado College. To them all i extend my most sincere
thanks. At a later stage, much of the manuscript was read by three people
who provided many helpful suggestions for its improvement. Theyare Prof.
C. Desoer of the University of California, Berkeley; Prof. D. J. Lewis of the
Universityof Michigan; and Prof.T. L. Wade of Florida State University. I am
sincerely grateful tothemforthisassistance. Themanuscriptin itsnear-final
form was read by Prof. D. J. Sterlingof Bowdoin College, and lwishtoexpress
my gratitude to him for his very useful suggestions; I am confident that the
book is better because of his pertinent commentary. It is understood,
of course, that even though these people have seen part or all of the manu-
script, their reading of it does not imply any sort of endorsement; any errors
or shortcomings in the bookare mine alone. My longtime friend, Prof. W. S.
Cannon of Presbyterian College, was of very considerable help in collecting
Readings from the American Mathematical Monthly. My colleague, Prof.
Anne Bode, has been a mostgraciousconsultanton matterswhich havearisen
from time to time, and her generous assistance continued throughout the
final stagesofthe manuscript. Twoverygood graduate students, EddySmet
and Jawaid Rizvi, came to my aid at a very busy time and used a portion of a
vacation period to supply mewith answersforthefirstgroupingsof problems.
Theirfurtherhelpin readingthegalleyswasinestimable,andthe bookismuch
closer to being free of errors than it would otherwise have been. I offer my
mostwarm and cordial thanks to them all.
JOHN T. MOORE