Elementary linear algebra

954 Pages·2016·3.57 MB·English

by  Cooperstein B.

ii
Bruce N. Cooperstein
Elementary Linear Algebra: An
eTextbook
Email:[email protected]
(cid:13)c2010,AllRightsReserved.Version2.0
February24,2016
Contents
1 LinearEquations 3
1.1 LinearEquationsandTheirSolution . . . . . . . . . . . . . . . . . . 3
1.2 MatricesandEchelonForms . . . . . . . . . . . . . . . . . . . . . . 32
1.3 HowtoUseit: ApplicationsofLinearSystems . . . . . . . . . . . . 62
2 TheVectorSpaceRn 91
2.1 IntroductiontoVectors: LinearGeometry . . . . . . . . . . . . . . . 91
2.2 VectorsandtheSpaceRn . . . . . . . . . . . . . . . . . . . . . . . . 116
2.3 TheSpanofaSequenceofVectors . . . . . . . . . . . . . . . . . . . 140
2.4 LinearindependenceinRn . . . . . . . . . . . . . . . . . . . . . . . 165
2.5 SubspacesandBasesofRn . . . . . . . . . . . . . . . . . . . . . . . 194
2.6 TheDotProductinRn . . . . . . . . . . . . . . . . . . . . . . . . . 223
3 MatrixAlgebra 251
3.1 IntroductiontoLinearTransformationsandMatrixMultiplication . . 251
3.2 TheProductofaMatrixandaVector. . . . . . . . . . . . . . . . . . 273
3.3 MatrixAdditionandMultiplication. . . . . . . . . . . . . . . . . . . 294
3.4 InvertibleMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
3.5 ElementaryMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . 340
3.6 TheLUFactorization . . . . . . . . . . . . . . . . . . . . . . . . . . 361
3.7 HowtoUseIt: ApplicationsofMatrixMultiplication . . . . . . . . . 379
4 Determinants 403
4.1 IntroductiontoDeterminants . . . . . . . . . . . . . . . . . . . . . . 403
4.2 PropertiesofDeterminants . . . . . . . . . . . . . . . . . . . . . . . 421
4.3 TheAdjointofaMatrixandCramer’sRule . . . . . . . . . . . . . . 447
5 AbstractVectorSpaces 461
5.1 IntroductiontoAbstractVectorSpaces . . . . . . . . . . . . . . . . . 461
5.2 SpanandIndependenceinVectorSpaces. . . . . . . . . . . . . . . . 478
5.3 Dimensionofafinitegeneratedvectorspace . . . . . . . . . . . . . . 510
5.4 Coordinatevectorsandchangeofbasis. . . . . . . . . . . . . . . . . 529
5.5 RankandNullityofaMatrix . . . . . . . . . . . . . . . . . . . . . . 551
iv CONTENTS
5.6 ComplexVectorSpaces . . . . . . . . . . . . . . . . . . . . . . . . 573
5.7 VectorSpacesOverFiniteFields . . . . . . . . . . . . . . . . . . . . 600
5.8 HowtoUseit: ErrorCorrectingCodes . . . . . . . . . . . . . . . . . 615
6 LinearTransformations 635
6.1 IntroductiontoLinearTransformationsonAbstractVectorSpaces . . 635
6.2 RangeandKernelofaLinearTransformation . . . . . . . . . . . . . 656
6.3 MatrixofaLinearTransformation . . . . . . . . . . . . . . . . . . . 680
7 EigenvaluesandEigenvectors 703
7.1 IntroductiontoEigenvaluesandEigenvectors . . . . . . . . . . . . . 703
7.2 DiagonalizationofMatrices . . . . . . . . . . . . . . . . . . . . . . 727
7.3 ComplexEigenvaluesofRealMatrices . . . . . . . . . . . . . . . . . 753
7.4 HowtoUseIt: ApplicationsofEigenvaluesandEigenvectors . . . . . 780
8 OrthogonalityinRn 803
8.1 OrthogonalandOrthonormalSetsinRn . . . . . . . . . . . . . . . . 803
8.2 TheGram-SchmidtProcessandQR-Factorization . . . . . . . . . . 824
8.3 OrthogonalComplementsandProjections . . . . . . . . . . . . . . . 843
8.4 DiagonalizationofRealSymmetricMatrices. . . . . . . . . . . . . . 869
8.5 QuadraticForms,ConicSectionsandQuadraticSurfaces . . . . . . . 895
8.6 HowtoUseIt: LeastSquaresApproximation . . . . . . . . . . . . . 932
Introduction: How to Use This
Book
This book will probably be unlike any textbook you have ever used and it has been
specificallywrittentofacilitateyourlearninginelementarylinearalgebra. Ithasbeen
my experience over more than forty five years of teaching linear algebra that student
successishighlycorrelatedwithmasteryofthedefinitionsandconceptsofwhichthere
arecertainlymorethenfiftyandverylikelyasmanyasonehundred. Inatypicalbook
if there is a term or concept that you are not completely familiar with you would go
to the index and find the page where it is defined. In a math class such as this, often
whenyoudothatyouwillfindthatthedefinitionmakesuseofotherconceptswhich
you have not mastered which would send you back to the index or, more likely, give
up. Thisdigitaltextbookmakesiteasytostudyandmasterthedefinitionsintwoways.
First,everyinstanceofafundamentalconceptislinkedtoitsdefinitionsoyoucanlook
itupjustbyclickingandthenyoucanreturntoyouroriginalplace.Second,atthevery
beginningofeverysectionisasubsection,“AmIReadyforThisMaterial”youwill
findlistedalltheconceptsthathavebeenpreviouslyintroducedandusedinthesection.
Youwillalsofindashortquizwhichtestswhetheryouhavemasteredthemethodsfrom
previoussectionswhichwillbeusedinthecurrentone. Thequizislinkedtosolutions
andthese,inturn,arelinkedtoexplanationsofhowyoudothatkindofexercise.
Subsequent to the “readiness subsection” is a subsection referred to as “New Con-
cepts”. Hereyouarealertedtothedefinitionswhichwillbeintroducedinthesection
andtheyarelinkedtotheplacewheretheyareintroduced.
Following is the “heart” of the section, a subsection referred to as “Theory: Why It
Works.” Here the new concepts are introduced and theorems about them are proved.
Thetheoremsarethenusedthroughoutthetexttoproveothertheorems. Aparticular
featureofthebookisthatanycitationofatheoremtoproveanothertheoremislinked
backtoitsoriginalstatementanditsproof.
Next comes a feature that is missing from most elementary linear algebra textbooks
tothefrustrationofmanystudents. First,asubsectionentitled“WhatYouCanNow
Do.”Thiswillalertyoutothenewtypesofexercisesthatyoucansolvebasedonthe
theoremsprovedintheTheorysubsection. Thenthereisasubsection,“Method: How
2 CONTENTS
todoit”whichprovidesexplicitalgorithms,essentiallyrecipes,forsolvingtheseex-
ercises. Ineachinstance,theseareillustratedwithmultipleexamples. Itistheabsence
of these algorithms from most expensive hard copy textbooks that sends students to
Amazon.com to spend an addition $15-$20 to purchase Schaum’s Outline of Linear
Algebra in order to figure out how to get the homework done. Note: If you are only
reading the Theory subsection you may get the impression that the book is bereft of
examples. You have to go to the Method subsection and there you will find multiple
examples,farmorethaninanyothertext.
Finally,thesectionendswith“Exercises”thataredoneusingthemethodsintroduced
(andaregenerallylinkedtothemethodsinordertoprimeyouaboutwhattouseand
howtoproceedaswellas“ChallengeExercises(CE)”orproblemsthatrequireyou
todosomerealmathematicalreasoningmakinguseofthetheoremsthatwereproved
witheachCElinkedtothetheoremortheoremsthatyouneedtocite.
Hopefully this text eases you way through elementary linear algebra. Typically text-
booksarewrittenforprofessors,notstudents,becauseprofessorschoosethebookand
studentsthenhavetobuyitandwhatprofessorswantinabookisoftenverydifferent
thenwhatastudentwantsinabook,especiallywhenthemajorityofstudentsarenot
inthecourseinordertobecomemathematicsmajorbutratherarestudyingengineer-
ing,computerscience,physics,economicsorsomeothersubject. Iftherearefeatures
[email protected]. Likewise,ifthereareportions
thatareunclearorconfusing,alsoletmeknowsinceIseethisasacontinuousproject
ofrefinementandimprovement.
BruceCooperstein
Professor,DepartmentofMathematics
UniversityofCalifornia,SantaCruz
Chapter 1
Linear Equations
1.1. Linear Equations and Their Solution
In this section we review the concepts of a linear equation and linear system. We
develop systematic methods for determining when a linear system has a solution and
forfindingthegeneralsolution. Belowisaguidetowhatyoufindinthissection.
AmIReadyforThisMaterial
ReadinessQuiz
NewConcepts
Theory(WhyItWorks)
WhatYouCanNowDo
Method(HowToDoIt)
Exercises
ChallengeExercises(Problems)
4 Chapter1.LinearEquations
Am I Ready for This Material
To be prepared for linear algebra you must have a complete mastery of pre-calculus,
some mathematical sophistication, some experience with proofs and the willingness
to work hard. More specifically, at the very least you need to be comfortable with
algebraic notation, in particular, the use of literals in equations to represent arbitrary
numbers as well as the application of subscripts. You should also be able to quickly
solveasinglelinearequationinonevariableaswellasasystemoftwolinearequations
intwovariables.
Youwillalsoneedsomemathematicalsophisticationinordertofollowthearguments
oftheproofsandevenattemptsomemathematicalreasoningofyourown.
The most important requirement is that you are willing to perspire, that is, put in the
necessary effort. Each section will introduce many new concepts and these have to
beunderstoodandmasteredbeforemovingon. “Kindof”understandingwillnotdo.
Nor will an “intuitive” understanding be sufficient - you will be often asked to fol-
lowdeductiveargumentsandevendosomeproofsyourself. Thesedependonadeep
and genuine understanding of the concepts, and there are lots of them, certainly over
fifty major definitions. So, its important to keep up and to periodically return to the
definitions and make sure that you truly understand them. With that in mind let me
sharewithyoumythree“axioms”oftimemanagementdevelopedoverfortyyearsof
teaching(andputtingtwosonsthroughuniversity):
◦Howevermuchtimeyouthinkyouhave,youalwayshaveless.
◦Howevermuchtimeyouthinkitwilltake,itwillalwaystakemore.
◦Somethingunexpectedwillcomeup.
Linearalgebraisabeautifulsubjectandatthesametimeaccessibletofirstandsecond
yearmathematicsstudents. Ihopethisbookconveystheeleganceofthesubjectandis
usefultoyourlearning. Goodluck.
Beforegoingontothematerialofthissectionyoushouldtakethequizheretoseeif
youhavetheminimummathematicalskillstosucceedinlinearalgebra.
Quiz
1. Solvetheequation
3(x−2)+7−2(x+4)=x−11
2. Solvethelinearequation:
1.1.LinearEquationsandTheirSolution 5
4(x−2)−2(x−4)=5(x−4)−4(x−5)
3. Solvethelinearequation:
3(2−x)−2(3−x)=4(3−x)−3(4−x)
4. Solvesystemoftwoequationsintwounknowns:
2x + 3y = 4
−3x + y = 5
QuizSolutions
New Concepts
linearequation
solutionoflinearequation
solutionsetofalinearequation
equivalentlinearequations
standardformofalinearequation
homogeneouslinearequation
inhomogeneouslinearequation
leadingtermlinearequationinstandardform
leadingvariableofalinearequationinstandardform
freevariablesofalinearequationinstandardform
linearsystem
constantsofalinearsystem
coefficientsofalinearsystem
homogenouslinearsystem
inhomogenouslinearsystem
solutionofalinearsystem
solutionsetofalinearsystem
6 Chapter1.LinearEquations
consistentlinearsystem
inconsistentlinearsystem
thetrivialsolutiontoahomogeneouslinearsystem
non-trivialsolutiontoahomogeneouslinearsystem
equivalentlinearsystems
echelonformforalinearsystem
leadingvariable
freevariable
elementaryequationoperation
Theory (Why It Works)
Beforegettingstartedabriefwordaboutanimportantconvention. Thisbookwillbe
dealing with particular types of equations, linear equations, which we define imme-
diately below. Equations involve “variables”. Usually these are real variables which
means that we can substitute real numbers for them and the result will be a state-
ment about the equality of two numbers. When there are just a few variables, two
or three or possibly four, we will typically use letters at the end of the alphabet, for
examplex,y fortwo, x,y,z whentherearethreevariables, andsometimesw,x,y,z
when there are four variables. When there are more than a few we will use a sin-
gle letter but “subscript” the letter with positive whole numbers starting at one, for
example, x ,x ,x ,x ,x ,... The subscript of such a variable is called its “index.”
1 2 3 4 5
We will also consider our variables to be ordered. The ordering is the natural one
for x ,x ,x ,x ,... while for smaller sets of variables we will use the alphabetical
1 2 3 4
ordering: x,yfortwovariables,x,y,zforthreeandw,x,y,zforfour.
Wenowbeginourdiscussionwithadefinitionthatisprobablyfamiliarfromprevious
mathematicsyouhavestudied:
Definition1.1. Alinearequationisanequationoftheform
b x +b x +···+b x +c=d x +d x +···+d x +e (1.1)
1 1 2 2 n n 1 1 2 2 n n
In Equation (1.1) the variables are x ,x ,...,x . The b ,d stand for (arbitrary but
1 2 n i i
fixed)realnumbersandarethecoefficientsofthevariablesandc,eare(arbitrarybut
fixed)realnumberswhicharetheconstantsoftheequation.