Linear Algebra Abridged

145 Pages·2016·1.24 MB·English

Linear Algebra Abridged
Sheldon Axler
This file is generated from Linear Algebra Done Right (third edition) by
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13March2016
©2015
Contents
1 VectorSpaces 1
1.A Rn andCn 2
ComplexNumbers 2
Lists 4
Fn 4
DigressiononFields 7
1.B DefinitionofVectorSpace 8
1.C Subspaces 11
SumsofSubspaces 12
DirectSums 13
2 Finite-DimensionalVectorSpaces 14
2.A SpanandLinearIndependence 15
LinearCombinationsandSpan 15
LinearIndependence 17
2.B Bases 19
2.C Dimension 21
3 LinearMaps 23
3.A TheVectorSpaceofLinearMaps 24
DefinitionandExamplesofLinearMaps 24
AlgebraicOperationsonL.V;W/ 24
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Contents vii
3.B NullSpacesandRanges 26
NullSpaceandInjectivity 26
RangeandSurjectivity 27
FundamentalTheoremofLinearMaps 28
3.C Matrices 29
RepresentingaLinearMapbyaMatrix 29
AdditionandScalarMultiplicationofMatrices 30
MatrixMultiplication 32
3.D InvertibilityandIsomorphicVectorSpaces 35
InvertibleLinearMaps 35
IsomorphicVectorSpaces 36
LinearMapsThoughtofasMatrixMultiplication 37
Operators 39
3.E ProductsandQuotientsofVectorSpaces 39
ProductsofVectorSpaces 39
ProductsandDirectSums 40
QuotientsofVectorSpaces 41
3.F Duality 44
TheDualSpaceandtheDualMap 44
TheNullSpaceandRangeoftheDualofaLinearMap 45
TheMatrixoftheDualofaLinearMap 47
TheRankofaMatrix 48
4 Polynomials 49
ComplexConjugateandAbsoluteValue 50
UniquenessofCoefficientsforPolynomials 51
TheDivisionAlgorithmforPolynomials 52
ZerosofPolynomials 52
FactorizationofPolynomialsoverC 53
FactorizationofPolynomialsoverR 55
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viii Contents
5 Eigenvalues,Eigenvectors,andInvariantSubspaces 57
5.A InvariantSubspaces 58
EigenvaluesandEigenvectors 59
RestrictionandQuotientOperators 60
5.B EigenvectorsandUpper-TriangularMatrices 61
PolynomialsAppliedtoOperators 61
ExistenceofEigenvalues 62
Upper-TriangularMatrices 63
5.C EigenspacesandDiagonalMatrices 66
6 InnerProductSpaces 69
6.A InnerProductsandNorms 70
InnerProducts 70
Norms 73
6.B OrthonormalBases 76
LinearFunctionalsonInnerProductSpaces 79
6.C OrthogonalComplementsandMinimizationProblems 80
OrthogonalComplements 80
MinimizationProblems 82
7 OperatorsonInnerProductSpaces 84
7.A Self-AdjointandNormalOperators 85
Adjoints 85
Self-AdjointOperators 86
NormalOperators 88
7.B TheSpectralTheorem 89
TheComplexSpectralTheorem 89
TheRealSpectralTheorem 90
7.C PositiveOperatorsandIsometries 92
PositiveOperators 92
Isometries 93
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Contents ix
7.D PolarDecompositionandSingularValueDecomposition 94
PolarDecomposition 94
SingularValueDecomposition 95
8 OperatorsonComplexVectorSpaces 97
8.A GeneralizedEigenvectorsandNilpotentOperators 98
NullSpacesofPowersofanOperator 98
GeneralizedEigenvectors 99
NilpotentOperators 100
8.B DecompositionofanOperator 101
DescriptionofOperatorsonComplexVectorSpaces 101
MultiplicityofanEigenvalue 102
BlockDiagonalMatrices 103
SquareRoots 104
8.C CharacteristicandMinimalPolynomials 105
TheCayley–HamiltonTheorem 105
TheMinimalPolynomial 106
8.D JordanForm 107
9 OperatorsonRealVectorSpaces 109
9.A Complexification 110
ComplexificationofaVectorSpace 110
ComplexificationofanOperator 111
TheMinimalPolynomialoftheComplexification 112
EigenvaluesoftheComplexification 112
CharacteristicPolynomialoftheComplexification 114
9.B OperatorsonRealInnerProductSpaces 115
NormalOperatorsonRealInnerProductSpaces 115
IsometriesonRealInnerProductSpaces 117
10 TraceandDeterminant 118
10.A Trace 119
ChangeofBasis 119
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x Contents
Trace: AConnectionBetweenOperatorsandMatrices 121
10.B Determinant 123
DeterminantofanOperator 123
DeterminantofaMatrix 125
TheSignoftheDeterminant 129
Volume 130
PhotoCredits 135
Index 136
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CHAPTER
1
RenéDescartesexplaininghis
worktoQueenChristinaof
Sweden. Vectorspacesarea
generalizationofthe
descriptionofaplaneusing
twocoordinates,aspublished
byDescartesin1637.
Vector Spaces
Linearalgebraisthestudyoflinearmapsonfinite-dimensionalvectorspaces.
Eventuallywewilllearnwhatallthesetermsmean. Inthischapterwewill
definevectorspacesanddiscusstheirelementaryproperties.
In linear algebra, better theorems and more insight emerge if complex
numbers are investigated along with real numbers. Thus we will begin by
introducingthecomplexnumbersandtheirbasicproperties.
We will generalize the examples of a plane and ordinary space to Rn
andCn,whichwethenwillgeneralizetothenotionofavectorspace. The
elementarypropertiesofavectorspacewillalreadyseemfamiliartoyou.
Thenournexttopicwillbesubspaces,whichplayaroleforvectorspaces
analogoustotheroleplayedbysubsetsforsets. Finally,wewilllookatsums
ofsubspaces(analogoustounionsofsubsets)anddirectsumsofsubspaces
(analogoustounionsofdisjointsets).
LEARNING OBJECTIVES FOR THIS CHAPTER
basicpropertiesofthecomplexnumbers
Rn andCn
vectorspaces
subspaces
sumsanddirectsumsofsubspaces
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2 CHAPTER 1 VectorSpaces
1.A Rn and Cn
ComplexNumbers
1.1 Definition complexnumbers
(cid:15) Acomplexnumberisanorderedpair.a;b/,wherea;b 2 R,but
wewillwritethisasaCbi.
(cid:15) ThesetofallcomplexnumbersisdenotedbyC:
C D faCbi W a;b 2 Rg:
(cid:15) AdditionandmultiplicationonC aredefinedby
.aCbi/C.c Cdi/ D .aCc/C.bCd/i;
.aCbi/.c Cdi/ D .ac (cid:0)bd/C.ad Cbc/iI
herea;b;c;d 2 R.
Ifa 2 R,weidentifyaC0i withtherealnumbera. Thuswecanthink
ofR asasubsetofC. Wealsousuallywrite0Cbi asjustbi,andweusually
write0C1i asjusti.
Using multiplication as defined above, you should verify that i2 D (cid:0)1.
Donotmemorizetheformulafortheproductoftwocomplexnumbers;you
canalwaysrederiveitbyrecallingthati2 D (cid:0)1andthenusingtheusualrules
ofarithmetic(asgivenby1.3).
1.3 Properties of complex arithmetic
commutativity
˛Cˇ D ˇC˛ and˛ˇ D ˇ˛ forall˛;ˇ 2 C;
associativity
.˛Cˇ/C(cid:21) D ˛C.ˇC(cid:21)/and.˛ˇ/(cid:21) D ˛.ˇ(cid:21)/forall˛;ˇ;(cid:21) 2 C;
identities
(cid:21)C0 D (cid:21)and(cid:21)1 D (cid:21)forall(cid:21) 2 C;
additiveinverse
forevery˛ 2 C,thereexistsauniqueˇ 2 C suchthat˛Cˇ D 0;
multiplicativeinverse
forevery˛ 2 C with˛ ¤ 0,thereexistsauniqueˇ 2 C suchthat
˛ˇ D 1;
distributiveproperty
(cid:21).˛Cˇ/ D (cid:21)˛C(cid:21)ˇ forall(cid:21);˛;ˇ 2 C.
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SECTION 1.A RnandCn 3
The properties above are proved using the familiar properties of real
numbersandthedefinitionsofcomplexadditionandmultiplication.
1.5 Definition (cid:0)˛,subtraction,1=˛,division
Let˛;ˇ 2 C.
(cid:15) Let (cid:0)˛ denote the additive inverse of ˛. Thus (cid:0)˛ is the unique
complexnumbersuchthat
˛C.(cid:0)˛/ D 0:
(cid:15) SubtractiononC isdefinedby
ˇ(cid:0)˛ D ˇC.(cid:0)˛/:
(cid:15) For˛ ¤ 0,let1=˛denotethemultiplicativeinverseof˛. Thus1=˛
istheuniquecomplexnumbersuchthat
˛.1=˛/ D 1:
(cid:15) DivisiononC isdefinedby
ˇ=˛ D ˇ.1=˛/:
So that we can conveniently make definitions and prove theorems that
applytobothrealandcomplexnumbers,weadoptthefollowingnotation:
1.6 Notation F
Throughoutthisbook,F standsforeitherR orC.
ThusifweproveatheoreminvolvingF,wewillknowthatitholdswhenF is
replacedwithR andwhenF isreplacedwithC.
Elements of F are called scalars. The word “scalar”, a fancy word for
“number”,isoftenusedwhenwewanttoemphasizethatanobjectisanumber,
asopposedtoavector(vectorswillbedefinedsoon).
For˛ 2 F andmapositiveinteger,wedefine˛m todenotetheproductof
˛ withitselfmtimes:
˛m D ˛(cid:1)(cid:1)(cid:1)˛:
„ƒ‚…
mtimes
Clearly.˛m/n D ˛mn and.˛ˇ/m D ˛mˇm forall˛;ˇ 2 F andallpositive
integersm;n.
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4 CHAPTER 1 VectorSpaces
Lists
TogeneralizeR2 andR3 tohigherdimensions,wefirstneedtodiscussthe
conceptoflists.
1.8 Definition list,length
Suppose n is a nonnegative integer. A list of length n is an ordered
collection of n elements (which might be numbers, other lists, or more
abstractentities)separatedbycommasandsurroundedbyparentheses. A
listoflengthnlookslikethis:
.x ;:::;x /:
1 n
Twolistsareequalifandonlyiftheyhavethesamelengthandthesame
elementsinthesameorder.
Thusalistoflength2isanorderedpair,andalistoflength3isanordered
triple.
Sometimes we will use the word list without specifying its length. Re-
member, however, that by definition each list has a finite length that is a
nonnegativeinteger. Thusanobjectthatlookslike
.x ;x ;:::/;
1 2
whichmightbesaidtohaveinfinitelength,isnotalist.
Alistoflength0lookslikethis: ./. Weconsidersuchanobjecttobea
listsothatsomeofourtheoremswillnothavetrivialexceptions.
Listsdifferfromsetsintwoways: inlists,ordermattersandrepetitions
havemeaning;insets,orderandrepetitionsareirrelevant.
Fn
To define the higher-dimensional analogues of R2 and R3, we will simply
replace R with F (which equals R or C) and replace the 2 or 3 with an
arbitrarypositiveinteger. Specifically,fixapositiveintegernfortherestof
thissection.
1.10 Definition Fn
Fn isthesetofalllistsoflengthnofelementsofF:
Fn D f.x ;:::;x / W x 2 F forj D 1;:::;ng:
1 n j
For .x ;:::;x / 2 Fn and j 2 f1;:::;ng, we say that x is the jth
1 n j
coordinateof.x ;:::;x /.
1 n
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