eBook Mathematics - Linear Algebra Book.pdf - Linear algebra - Kuya

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.

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

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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