Hefferson Jim - Linear Algebra.pdf

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Linear Algebra
Linear Algebra
JimHefferon
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Notation
¯ ¯ :::g setof...suchthat...
h:::i sequence;likeasetbutordermatters
V;W;U vectorspaces
~v;~w vectors
~ 0, ~ 0 V zerovector,zerovectorof V
B;D bases
E n = h~e 1 ;:::;~e n i standardbasisforR n
~ ¯; ~ ± basisvectors
Rep B ( ~v )matrixrepresentingthevector
P n setof n -thdegreepolynomials
M n£m setof n£m matrices
[ S ] spanoftheset S
M©N directsumofsubspaces
V » = W isomorphicspaces
h;g homomorphisms,linearmaps
H;G matrices
t;s transformations;mapsfromaspacetoitself
T;S squarematrices
Rep B;D ( h )matrixrepresentingthemap h
h i;j matrixentryfromrow i ,column j
jTj determinantofthematrix T
R ( h ) ;N ( h ) rangespaceandnullspaceofthemap h
R 1 ( h ) ;N 1 ( h ) generalizedrangespaceandnullspace
R realnumbers
N naturalnumbers: f 0 ; 1 ; 2 ;:::g
C complexnumbers
f:::
LowercaseGreekalphabet
name charactername charactername character
alpha ® iota rho ½
beta ¯ kappa · sigma ¾
gamma ° lambda ¸ tau ¿
delta ± mu ¹ upsilon À
epsilon ² nu º phi Á
zeta ³ xi » chi Â
eta ´ omicron o psi Ã
theta µ pi ¼ omega !
Cover.ThisisCramer’sRuleforthesystem x +2 y =6,3 x + y =8.Thesizeofthe
firstboxisthedeterminantshown(theabsolutevalueofthesizeisthearea).The
sizeofthesecondboxis x timesthat,andequalsthesizeofthefinalbox.Hence, x
isthefinaldeterminantdividedbythefirstdeterminant.
133729284.003.png 133729284.004.png 133729284.005.png
Preface
Inmostmathematicsprogramslinearalgebracomesinthefirstorsecondyear,
followingoralongwithatleastonecourseincalculus.Whilethelocation
ofthiscourseisstable,latelythecontenthasbeenunderdiscussion.Some
instructorshaveexperimentedwithvaryingthetraditionaltopicsandothers
havetriedcoursesfocusedonapplicationsoroncomputers.Despitethishealthy
debate,mostinstructorsarestillconvinced,Ithink,thattherightcorematerial
isvectorspaces,linearmaps,determinants,andeigenvaluesandeigenvectors.
Applicationsandcodehaveaparttoplay,butthethemesofthecourseshould
remainunchanged.
Notthatallisfinewiththetraditionalcourse.Manyofusbelievethatthe
standardtexttypecoulddowithachange.Introductorytextshavetraditionally
startedwithextensivecomputationsoflinearreduction,matrixmultiplication,
anddeterminants,whichtakeuphalfofthecourse.Then,whenvectorspaces
andlinearmapsfinallyappearanddefinitionsandproofsstart,thenatureofthe
coursetakesasuddenturn.Thecomputationdrillwasthereinthepastbecause,
asfuturepractitioners,studentsneededtobefastandaccurate.Butthathas
changed.Beingawhizat5 £ 5determinantsjustisn’timportantanymore.
Instead,theavailabilityofcomputersgivesusanopportunitytomovetoward
afocusonconcepts.
Thisisanopportunitythatweshouldseize.Thecoursesatthestartofmost
mathematicsprogramsworkathavingstudentsapplyformulasandalgorithms.
Latercoursesaskformathematicalmaturity:reasoningskillsthataredeveloped
enoughtofollowdi®erenttypesofarguments,afamiliaritywiththethemes
thatunderlymanymathematicalinvestigationslikeelementarysetandfunction
facts,andanabilitytodosomeindependentreadingandthinking.Wheredo
weworkonthetransition?
Linearalgebraisanidealspot.Itcomesearlyinaprogramsothatprogress
madeherepayso®later.But,itisalsoplacedfarenoughintoaprogram
thatthestudentsareseriousaboutmathematics,oftenmajorsandminors.
Thematerialisstraightforward,elegant,andaccessible.Thereareavarietyof
argumentstyles—proofsbycontradiction,ifandonlyifstatements,andproofs
byinduction,forinstance—andexamplesareplentiful.
Thegoalofthistextis,alongwiththepresentationofundergraduatelinear
algebra,tohelpaninstructorraisethestudents’levelofmathematicalsophis-
tication.Mostofthedi®erencesbetweenthisbookandothersfollowstraight
fromthatgoal.
Oneconsequenceofthisgoalofdevelopmentisthat,unlikeinmanycompu-
tationaltexts,alloftheresultshereareproved.Ontheotherhand,incontrast
withmoreabstracttexts,manyexamplesaregiven,andtheyareoftenquite
detailed.
Anotherconsequenceofthegoalisthatwhilewestartwithacomputational
topic,linearreduction,fromthefirstwedomorethanjustcompute.The
iii
solutionoflinearsystemsisdonequicklybutcompletely,provingeverything,
allthewaythroughtheuniquenessofreducedechelonform.And,rightinthis
firstchaptertheopportunityistakentopresentafewinductionproofs,where
theargumentsarejustverificationsofdetails,sothatwheninductionisneeded
later(e.g.,toprovethatallbasesofafinitedimensionalvectorspacehavethe
samenumberofmembers)itwillbefamiliar.
Stillanotherconsequenceofthegoalofdevelopmentisthatthesecondchap-
terstarts(usingthelinearsystemsworkasmotivation)withthedefinitionofa
realvectorspace.Thistypicallyoccursbytheendofthethirdweek.Wedonot
stoptointroducematrixmultiplicationanddeterminantsasrotecomputations.
Instead,thosetopicsappearnaturallyinthedevelopment,afterthedefinition
oflinearmaps.
Throughoutthebookthepresentationstressesmotivationandnaturalness.
Anexampleisthethirdchapter,onlinearmaps.Itdoesnotbeginwiththe
definitionofahomomorphism,asisthecaseinotherbooks,butwiththat
ofanisomorphism.That’sbecauseisomorphismiseasilymotivatedbythe
observationthatsomespacesarejustlikeeachother.Afterthat,thenext
sectiontakesthereasonablestepofdefininghomomorphismsbyisolatingthe
operation-preservationidea.Somemathematicalslicknessislost,butitisin
returnforalargegaininsensibilitytostudents.
Havingextensivemotivationinthetextalsohelpswithtimepressures.I
askstudentsto,beforeeachclass,lookaheadinthebook.Theyfollowthe
classworkbetterbecausetheyhavesomepriorexposuretothematerial.For
example,Icanstartthelinearindependenceclasswiththedefinitionbecause
Iknowstudentshavesomeideaofwhatitisabout.Nobookcantakethe
placeofaninstructorbutahelpfulbookgivestheinstructormoreclasstime
forexamplesandquestions.
Muchofastudent’sprogresstakesplacewhiledoingtheexercises;theex-
erciseshereworkwiththerestofthetext.Besidescomputations,thereare
manyproofs.Ineachsubsectiontheyarespreadoveranapproachabilityrange,
fromsimplecheckstosomemuchmoreinvolvedarguments.Thereareevena
fewthatarechallengingpuzzlestakenfromvariousjournals,competitions,or
problemscollections(thesearemarkedwitha?;aspartofthefun,theorigi-
nalwordinghasbeenretainedasmuchaspossible).Intotal,theexercisesare
aimedtobothbuildanabilityat,andhelpstudentsexperiencethepleasureof,
doing mathematics.
Applications,andComputers.Thepointofviewtakenhere,thatlinear
algebraisaboutvectorspacesandlinearmaps,isnottakentotheexclusionof
allothers.Applicationsandtheroleofthecomputerareinteresting,important,
andvitalaspectsofthesubject.Consequently,everychaptercloseswithafew
applicationorcomputer-relatedtopics.Someoftheseare:networkflows,the
speedandaccuracyofcomputerlinearreductions,LeontiefInput/Outputanal-
ysis,dimensionalanalysis,Markovchains,votingparadoxes,analyticprojective
geometry,andsolvingdi®erenceequations.
Thesetopicsarebriefenoughtobedoneinaday’sclassortobegivenas
iv
independentprojectsforindividualsorsmallgroups.Mostsimplygiveareader
afeelforthesubject,discusshowlinearalgebracomesin,pointtosomefurther
reading,andgiveafewexercises.Ihavekepttheexpositionlivelyandgivenan
overallsenseofbreadthofapplication.Inshort,thesetopicsinvitereadersto
seeforthemselvesthatlinearalgebraisatoolthataprofessionalmusthave.
Forpeoplereadingthisbookontheirown. Theemphasishereon
motivationanddevelopmentmakethisbookagoodchoiceforself-study.But
whileaprofessionalinstructorcanjudgewhatpaceandtopicssuitaclass,
perhapsanindependentstudentwouldfindsomeadvicehelpful.Herearetwo
timetablesforasemester.Thefirstfocusesoncorematerial.
weekMonday Wednesday Friday
1One.I.1 One.I.1,2 One.I.2,3
2One.I.3 One.II.1 One.II.2
3One.III.1,2 One.III.2 Two.I.1
4Two.I.2 Two.II Two.III.1
5Two.III.1,2 Two.III.2 exam
6Two.III.2,3 Two.III.3 Three.I.1
7Three.I.2 Three.II.1 Three.II.2
8Three.II.2 Three.II.2 Three.III.1
9Three.III.1 Three.III.2 Three.IV.1,2
10Three.IV.2,3,4 Three.IV.4 exam
11Three.IV.4,Three.V.1Three.V.1,2 Four.I.1,2
12Four.I.3 Four.II Four.II
13Four.III.1 Five.I Five.II.1
14Five.II.2 Five.II.3 review
Thesecondtimetableismoreambitious(itpresupposesOne.II,theelementsof
vectors,usuallycoveredinthirdsemestercalculus).
weekMonday Wednesday Friday
1One.I.1 One.I.2 One.I.3
2One.I.3 One.III.1,2 One.III.2
3Two.I.1 Two.I.2 Two.II
4Two.III.1 Two.III.2 Two.III.3
5Two.III.4 Three.I.1 exam
6Three.I.2 Three.II.1 Three.II.2
7Three.III.1 Three.III.2 Three.IV.1,2
8Three.IV.2 Three.IV.3 Three.IV.4
9Three.V.1 Three.V.2 Three.VI.1
10Three.VI.2 Four.I.1 exam
11Four.I.2 Four.I.3 Four.I.4
12Four.II Four.II,Four.III.1 Four.III.2,3
13Five.II.1,2 Five.II.3 Five.III.1
14Five.III.2 Five.IV.1,2 Five.IV.2
Seethetableofcontentsforthetitlesofthesesubsections.
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