Vector Spaces of Finite Dimension

208 Pages·1966·9.254 MB·English

by  G. C. Shephard

UNIVERSITY MATHEMATICALTEXTS
GENERAL EDITORS
ALEXANDER C. AITKEN, D.sc., mus.
DANIEL E. RUTHERFORD, D.sc., DR.MATH.
ASSISTANT nn'on
IAIN T. ADAMSON, PHD.
32
VECTOR SPACES
OF FINITE DIMENSION
UNIVERSITY MATHEMATICAL TEXTS
—. DeterminantsandMatrices A.C.Aitken,D.Sc.,F.R.S.
N. StatisticalMathematics A.C.Aitken,D.Sc.,F.R.S.
J. Integration ..R P.Gillespie,Ph.D.
U
A. IntegrationofOrdinaryDrierentialEqualions E.L.Ince,D.Sc.
IIE VectorMethods ..D E.Rutherford,D.Sc.,Dr.Math.
Q. Theoryoquations.. H.W.Tumbull,F.R.S.
Q. FunctionsofaComplex Variable E.G.Phillips, M.A. M.Sc.
W
. Waves .. ...C A.Coulson, M.A.,D..,Sc F..RS.
W
. InfiniteSeries .. J.M.Hyslop,D.Sc.
. AnalyticalGeometryof
ThreeDimensions W.H.McCrea,Ph..D,F.R.S.
. Electricity :C.A.Coulson,M..,A D..,Sc F.R.S.
. PrajectioeGeometry T..E Faulkner,Ph.D.
. IntroductiontotheTheoryof
FiniteGroups ..W Ledermann,Ph.D. D.Sc.
. ClassicalMechanics D.E.Rutherford,D.Sc.,Dr.Math.
. PartialDiferentiation .. ..R P.Gillespie,Ph.D.
. VolumeandIntegral W.W.Rogosinskl,DrPhil.,F.R.S.
. TensorCalculus ... ... B.Spaln,Ph.D.
. German-EnglishMathematical
Vocabulary S.Macintyre,Ph.D.,andE.Witte, M.A.
SpecialFunctionsofMathematical
PhysicsandChemistry I.N.Sneddon,D.Sc.
. Topology E.M.Patterson,Ph.D.
. TheTheoryofOrdinary
DfirentialEquations ... J.C.Burkill,Sc.D.,F.R.S.
FluidDynamics D.E.Rutherford,D.50.,Dr.Math.
. SpecialRelativity W.Rindler,Ph.D.
. Real Variable J. M.Hyslop,D.Sc.
. RussianReaderin .
PureandAppliedMathematics P.H.Nldd1tch,Ph.D.
. Russian-EnglishMathematical
Vocabulary J.Burlak,M.Sc.,Ph.D.,aaK.Brooke,M.A.
. IntroductiontoFieldTheory IainT.Adamson,PhD
. NumberTheory .. ... J.Hunter,Ph.D.
. NumericalMethods. 1.Iteration,
ProgrammingandAlgebraicEquations. B Noble,D.Sc.
. NumericalMethods: 2. Diflerences,
IntegrationandDtfl'erentialEquations B.Noble,D.Sc.
. ElementaryAbstractAlgebra
E. M.Patterson,Ph..D,andD.E.Rutherford,D.Sc,Dr.Math.
. VectorSpacesofFiniteDimension G.C.Shephard,MHA,Ph.D.
. Magnetohydrodynamics .. A.Jeffrey,Ph.D.
VECTOR SPACES
of Finite Dimension
G. C. SHEPHARD
M.A.,Ph.D.
SeniorLecturerinMathematics
intheUniversityofBirmingham
OLIVER & BOYD
EDINBURGH AND LONDON
NEW YORK! INTERSCIENCE PUBLISHERS INC.
A DIVISION OF JOHN WILEY & SONS, INC.
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OLIVER AND BOYD L‘ID.,-1NDURGH
PREFACE
THISbookisbasedonacourseoflectures onlinearalgebra
given to second-year Honours students in the University
ofBirmingham.
Afamiliaritywiththemostelementaryparts ofabstract
algebra, together with the properties of matrices and
determinants, is assumed. These are summarised briefly
intheintroductorysection,afterwhichthefirstfivechapters
give a concise account ofthe theory offinite-dimensional
vectorspaces without theuse ofmatrices. In ChapterVI,
matricesareintroduced,andthemethodsalreadydeveloped
are applied to them.
Mostofthenotationsandterminologyusedarestandard.
We have borrowed the notation (xly) from quantum
theory for the inner product of two vectors (Chapter V)
and in Chapter II we have introduced a new term,implicit
rank, for a quantity which plays an important role in the
reduction oflinear transformations, but does not seem to
havepreviously been named.
Over 150 exercises areincludedin thetext. These vary
greatly in difliculty: some are simple questions which the
readershould be able to answerimmediately, whilst others
arebasedontopicswhichcouldnotbetreatedthroughlack
of space. As far as possible, the exercises have been
designed to test whether the reader has understood the
preceding material. With this in view, problems of a
purelycomputational naturehave beenexcluded.
The material presented here has been collected from
manysources,andIwouldliketoacknowledge,inparticular,
the following,books: Algébre Multilinéaire (Chapter III,
Book 2 of Ele'ments de Mathématique) by N. Bourbaki,
V
vi PREFACE
Linear Algebra by K. Hofiman and.R. Kunze, Introduction
to Linear Algebra by F. M. Stewart, Foundations ofLinear
Algebra by A. I. Mal’cev, Finite-dimensional VectorSpaces
by P. R. Halmos, as well as many books on matrixtheory
and its applications to linearalgebra.
FinallyIwouldliketoexpressmythankstothestudents
who originally suggested that I should write this book, to
Professor D. E. Rutherford, Dr I. T. Adamson, Dr W.
Jonsson, Dr A. H. M. Hoare and Mr Colin Campbell for
their helpful suggestions, and to Miss Gillian Rose for
typingthemanuscript.
G. C. SHEPHARD
BIRMINGHAM
June 1965
CONTENTS
PAGE
PREFACE V
SUMMARY OF SET THEORY AND ALGEBRA
0.1. Sets, relationsandfunctions l
0.2. Groups,ringsandfields 4
0.3. Matrices 6
CHAPTER I
VECTOR SPACES AND SUBSPACES
. Thedefinitionofavectorspace
. Subspaces 13
. Lineardependence,bases,dimension
. Sumsandintersectionsofsubspaces 25
CHAPTER II
LINEAR TRANSFORMATIONS
. Thedefinitionofalineartransformation
. Rankandnullity
. Sumsandscalarproductsoflineartransformations
. Compositionoflineartransformations
. Inverse:oflineartransformations
. Invariantsubspaces
. Projections
. Characteristicrootsandcharacteristicvectors
. Reductiontheorems
CHAPTER III
DUAL VECTOR SPACES ‘
. Linearfunctionalsandduality
. Annihilators
. Thedualofthedualspace
. Dualtransformations
vii
vill CONTENTS
CHAPTER IV
MULTILINEAR ALGEBRA
IAOI
. Bilinearfunctionals
. Tensorproducts 104
. Antisymmetricfunctionals
. Wedgeproducts 116
. Determinantsandcharacteristicequations 122
CHAPTER V
NORMS AND INNER PRODUCTS
. Norms 132
. Innerproducts 135
. Orthogonalcomplements 144
. Dualspacesandinnerproducts 146
. Adjointtransformations 151
. Isometries 154
. . Self-adjointtransformations 157
CHAPTER VI
COORDINATES AND MATRICES
. Coordinates 163
. Lineartransformationsintermsofcoordinates
. Changeofbases 176
. Canonicalformsformatrices 187
SOLUTIONS To EXERCISES 195
INDEX or DEPeons 197
SUMMARY OF SET THEORY AND ALGEBRA
§0.1. Sets,RelationsandFunctions
Upper-case letter A, B, C, ..., X, Y, Z will usually be
used for sets, and their elements will be denoted by lower-
case letters a, b, c, ..., x,y, z. The notation xeXmeans
thattheelementxbelongsto theset X, andwewrite
X={x1, ..., x,}
ifXis afinite setwith relements x1, ..., x,. The set with
no elementsiscalledthe emptyset. Theintersection XnY
oftwo sets Xand Yisthesetofall elementswhich belong
bothto Xandto Y. IfXnYistheemptyset, then Xand
Y are said to be disjoint. The union XuY of X and Y
is the set ofall elements whichbelong to Xor to Y(or to
both). Ifevery element of Xis an element of Y, we say
that Xisasubsetof Yandwrite XcY(thepossibilitythat
X= Y is not excluded). The cartosian product of two
sets, Xx Y, is defined as the set ofall ordered pairs (x, y)
with xeXandy6 Y. XxXis written X2, and similarly,
XxXx xX(the setofall orderedn—tuples(x1, ...,x") of
elements ofX)will bedenoted by X".
A relation p between two sets X and Y is defined as
any subset of Xx Y, i.e., it is a set of ordered pairs. If
(x, y)ep we mayalternatively write xpy and say that x
is related toy. IfX= Y, the relation pc2 is said to be
in the set X. The relation p in the set Xconsisting ofall
pairs (x, x)is called theidentity relation.
Ifp is arelationbetweensets Xand Y,thenthe inverse
l
2 VECTOR SPACES OF FINITE DIMENSION §0.1
relation p" between the sets Yand Xis defined astheset
ofall pairs (y, x) for which(x,y)ep.
A relation p in a set Xis called anequivalence relation
ifit satisfies the followingthreeconditions:
El. Itis reflexive, thatis, xpxfor all xeX.
E2. Itis symmetric, thatis, x1pxz ifand onlyifxs1.
E3. It is transitive. This means that ifxlpx2 and xs3,
then XIPX3.
Corresponding to each equivalence relation in a set X
there is apartition ofXinto disjointsubsets such thattwo
elements belong to the same subset ofthe partition ifand
only if they are related by the given equivalence relation,
or, more briefly, if they are equivalent. The subsets of
thepartitionarecalledequivalenceclasses. Theequivalence
classcontaininganelementxisdenotedby [x].
Another important type of relation between two sets
is a function f:Xx Y. This is a relation that satisfies
the property that each xeX is the first element of one
and only one pair (x, y)ef. It is convenient to regardf
as a mapping or transformation, that is, regard it as
associating with each element xeX, a unique element
yG Y. We writethis
f: X—+Y
and say thatfmaps Xinto Y. f(x) is used to denote the
image ofx, that is to say the second elementy ofthe pair
(x, y). Notice thatf signifies the function, whereas f(x)
is an element of Y. The set Xis called the domain off,
and the set of all elementsf(x), as x runs through X, is
therangeoff, convenientlydenoted byf(X). Iff(X) = Y,
fis said to be onto Y or surjective and ifeach element y
arises as the image of a unique element xeX, f is said
to be 1—1 or injective. Since a function is a relation, it