linear algebra book

319 Pages·2004·1.44 MB·English

Linear Algebra
Maths 270 c
(cid:13)
DavidA.SANTOS
Community College of Philadelphia: SPRING 2004
March 3, 2004 Revision

Contents
Preface v
1 Preliminaries 1
1.1 SetsandNotation. . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 PartitionsandEquivalenceRelations . . . . . . . . . . . . . . 5
1.3 BinaryOperations . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
n
1.5 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2 MatricesandMatrixOperations 25
2.1 TheAlgebraofMatrices . . . . . . . . . . . . . . . . . . . . . . 25
2.2 MatrixMultiplication . . . . . . . . . . . . . . . . . . . . . . . . 29
2.3 TraceandTranspose . . . . . . . . . . . . . . . . . . . . . . . . 36
2.4 SpecialMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5 MatrixInversion . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.6 BlockMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.7 RankofaMatrix. . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.8 RankandInvertibility . . . . . . . . . . . . . . . . . . . . . . . . 69
3 LinearEquations 79
3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.2 ExistenceofSolutions. . . . . . . . . . . . . . . . . . . . . . . . 84
3.3 ExamplesofLinearSystems . . . . . . . . . . . . . . . . . . . . 86
4 R2,R3 andRn 93
4.1 PointsandBi-pointsinR2 . . . . . . . . . . . . . . . . . . . . . 93
4.2 VectorsinR2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3 DotProductinR2 . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.4 LinesonthePlane . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.5 VectorsinR3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
iii
iv CONTENTS
4.6 PlanesandLinesinR3 . . . . . . . . . . . . . . . . . . . . . . . 126
4.7 Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
5 VectorSpaces 137
5.1 VectorSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.2 VectorSubspaces . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.3 LinearIndependence . . . . . . . . . . . . . . . . . . . . . . . 145
5.4 SpanningSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
5.5 Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.6 Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6 LinearTransformations 167
6.1 LinearTransformations . . . . . . . . . . . . . . . . . . . . . . . 167
6.2 KernelandImageofaLinearTransformation . . . . . . . . . 170
6.3 MatrixRepresentation . . . . . . . . . . . . . . . . . . . . . . . 174
7 Determinants 183
7.1 Permutations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.2 CycleNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
7.3 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
7.4 LaplaceExpansion . . . . . . . . . . . . . . . . . . . . . . . . . 205
7.5 DeterminantsandLinearSystems . . . . . . . . . . . . . . . . 213
8 EigenvaluesandEigenvectors 215
8.1 SimilarMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
8.2 EigenvaluesandEigenvectors . . . . . . . . . . . . . . . . . . 216
8.3 Diagonalisability . . . . . . . . . . . . . . . . . . . . . . . . . . 221
A SomeAnswersandHints 227
Preface
These notes started during the Spring of 2002, when John MAJEWICZ
andIeachtaughtasectionofLinearAlgebra. Iwouldliketothankhim
fornumeroussuggestionsonthewrittennotes.
The students of my class were: Craig BARIBAULT, Chun CAO, Jacky
CHAN, Pho DO, Keith HARMON, Nicholas SELVAGGI, Sanda SHWE, and
HuongVU.
John’s students were David HERNÁNDEZ, Adel JAILILI, Andrew KIM,
JongKIM,AbdelmounaimLAAYOUNI,AjuMATHEW,NikitaMORIN,Thomas
NEGRÓN,LatoyaROBINSON,andSaemSOEURN.
LinearAlgebraisoftenastudent’sfirstintroductiontoabstractmath-
ematics. LinearAlgebraiswellsuitedforthis,asithasanumberofbeau-
tifulbutelementaryandeasytoprovetheorems. Mypurposewiththese
notes is to introduce students to the concept of proof in a gentle man-
ner.
DavidA.SANTOS
ThisdocumentwaslastrevisedonMarch3,2004.
v
Thingsdone:
˚ RewrotethechapteronR2,R3,Rn
¸ Rewrotethesectiononcoordinates.
(cid:204) Addedmoreexercises.
Now,cananyone,fromanywhereintheworld,helpmewiththis?
Thingstodo:
˚ Writeasectiononbarycentresindimension2.
¸ ProveCeva’sandMenelaus’Theoremsusingbarycentres.
(cid:204) Writeasectiononisometries.
˝ Writeasectiononpositive-definitematrices.
˛ Writeasectiononquadrics.
ˇ Drawmorediagrams.
— Writeasectiononsumsofsubspaces.
1
Chapter
Preliminaries
1.1 Sets and Notation
1 Definition Wewillmeanbyasetacollectionofwelldefinedmembers
orelements.
2 Definition Thefollowingsetshavespecialsymbols.
N={0,1,2,3,...} denotesthesetofnaturalnumbers.
Z={...,−3,−2,−1,0,1,2,3,...} denotesthesetofintegers.
Q denotesthesetofrationalnumbers.
R denotesthesetofrealnumbers.
C denotesthesetofcomplexnumbers.
∅ denotestheemptyset.
3 Definition(Implications) The symbol = is read “implies”, and the
symbol isread“ifandonlyif.”
⇒
4 Exam⇐pl⇒e Prove that between any two rational numbers there is al-
waysarationalnumber.
Solution: Let(a,c) Z2,(b,d) (N\{0})2, a < c. Thenda<bc. Now
∈ ∈ b d
a a+c
ab+ad<ab+bc = a(b+d)<b(a+c) = < ,
b b+d
⇒ 1 ⇒
2 Chapter1
a+c c
da+dc<cb+cd = d(a+c)<c(b+d) = < ,
b+d d
a+c
whencetherationalnumb⇒er liesbetween a⇒and c.
b+d b d
!
Wecanalsoarguethattheaverageoftwodistinctnumbersliesbetweenthe
numbersandsoifr1 <r2arerationalnumbers,then r1+2r2 liesbetweenthem.
5 Definition LetAbeaset. IfabelongstothesetA,thenwewritea A,
∈
read “a is an element of A.” If a does not belong to the set A, we write
a A,read“aisnotanelementofA.”
6∈
6 Definition(Conjunction,Disjunction,andNegation) Thesymbol∨isread
“or” (disjunction), the symbol ∧ is read “and” (conjunction), and the
symbol¬isread“not.”
7 Definition(Quantifiers) Thesymbol isread“forall”(theuniversalquan-
∀
tifier),andthesymbol isread“thereexists”(theexistentialquantifier).
∃
Wehave
¬( x A,P(x)) ( A,¬P(x)) (1.1)
∀ ∈ ∃∈
⇐⇒
¬( A,P(x)) ( x A,¬P(x)) (1.2)
∃∈ ∀ ∈
8 Definition(Subset) If a A w⇐e⇒have a B, then we write A B,
∀ ∈ ∈ ⊆
whichweread“AisasubsetofB.”
Inparticular,noticethatforanysetA,∅ AandA A. Also
⊆ ⊆
N Z Q R C.
⊆ ⊆ ⊆ ⊆
!
A=B (A B)∧(B A).
⊆ ⊆
⇐⇒
9 Definition TheunionoftwosetsAandB,istheset
A B={x:(x A) ∨ (x B)}.
∪ ∈ ∈
Thisisread“AunionB.” Seefigure1.1.
SetsandNotation 3
10 Definition TheintersectionoftwosetsAandB,is
A B={x:(x A) ∧ (x B)}.
∩ ∈ ∈
Thisisread“AintersectionB.” Seefigure1.2.
11 Definition ThedifferenceoftwosetsAandB,is
A\B={x:(x A) ∧(x B)}.
∈ 6∈
Thisisread“AsetminusB.” Seefigure1.3.
A B A B A B
Figure1.1:A B Figure1.2:A B Figure1.3:A\B
∪ ∩
12 Example Provebymeansofsetinclusionthat
(A B) C=(A C) (B C).
∪ ∩ ∩ ∪ ∩
Solution: Wehave,
x (A B) C x (A B)∧x C
∈ ∪ ∩ ∈ ∪ ∈
(x A∨x B)∧x C
⇐⇒ ∈ ∈ ∈
(x A∧x C)∨(x B∧x C)
⇐⇒ ∈ ∈ ∈ ∈
(x A C)∨(x B C)
⇐⇒ ∈ ∩ ∈ ∩
x (A C) (B C),
⇐⇒ ∈ ∩ ∪ ∩
whichestablishestheequ⇐a⇒lity.
4 Chapter1
13 Definition LetA ,A ,...,A ,besets. TheCartesianProductofthese
1 2 n
nsetsisdefinedanddenotedby
A A A ={(a ,a ,...,a ):a A },
1 2 n 1 2 n k k
× ×···× ∈
that is, the set of all ordered n-tuples whose elements belong to the
givensets.
!
IntheparticularcasewhenalltheAk areequaltoasetA,wewrite
A1 A2 An =An.
× ×···×
Ifa Aandb Awewrite(a,b) A2.
∈ ∈ ∈
14 Definition Let x R. The absolute value of x—denoted by |x|—is
∈
definedby
−x if x<0,
|x|=

 x if x 0.
≥

Itfollowsfromthedefinitionthatforx R,
∈
−|x| x |x|. (1.3)
≤ ≤
t 0 = |x| t −t x t. (1.4)
≥ ≤ ≤ ≤
a⇒ R = ⇐√⇒a2 =|a|. (1.5)
∀ ∈
⇒
15Theorem(TriangleInequality) Let(a,b) R2. Then
∈
|a+b| |a|+|b|. (1.6)
≤
Proof From1.3,byaddition,
−|a| a |a|
≤ ≤
to
−|b| b |b|
≤ ≤
weobtain
−(|a|+|b|) a+b (|a|+|b|),
≤ ≤
whencethetheoremfollowsby1.4. q