Beezer Robert A. - A First Course In Linear Algebra.pdf - Linear Algebra - lemoniada91

AFirstCourseinLinearAlgebra

RobertA.Beezer

DepartmentofMathematicsandComputerScience

UniversityofPugetSound

Version0.70

January5,2006

c 2004,2005,2006

Permissionisgrantedtocopy,distributeand/ormodifythisdocumentunderthetermsof

theGNUFreeDocumentationLicense,Version1.2oranylaterversionpublishedbythe

FreeSoftwareFoundation;withtheInvariantSectionsbeing“Preface”,noFront-Cover

Texts,andnoBack-CoverTexts.Acopyofthelicenseisincludedinthesectionentitled

“GNUFreeDocumentationLicense”.

Mostrecentversioncanbefoundat http://linear.ups.edu/ .

Preface

Thistextbookisdesignedtoteachtheuniversitymathematicsstudentthebasicsof

thesubjectoflinearalgebra.Therearenoprerequisitesotherthanordinaryalgebra,

butitisprobablybestusedbyastudentwhohasthe“mathematicalmaturity”ofa

sophomoreorjunior.

Thetexthastwogoals:toteachthefundamentalconceptsandtechniquesofmatrix

algebraandabstractvectorspaces,andtoteachthetechniquesassociatedwithunder-

standingthedeﬁnitionsandtheoremsformingacoherentareaofmathematics.Sothere

isanemphasisonworkedexamplesofnontrivialsizeandonprovingtheoremscarefully.

Thisbookiscopyrighted.Thismeansthatgovernmentshavegrantedtheauthora

monopoly—theexclusiverighttocontrolthemakingofcopiesandderivativeworksfor

manyyears(toomanyyearsinsomecases).Italsogivesotherslimitedrights,generally

referredtoas“fairuse,”suchastherighttoquotesectionsinareviewwithoutseeking

permission.However,theauthorlicensesthisbooktoanyoneunderthetermsoftheGNU

FreeDocumentationLicense(GFDL),whichgivesyoumorerightsthanmostcopyrights.

Looselyspeaking,youmaymakeasmanycopiesasyoulikeatnocost,andyoumay

distributetheseunmodiﬁedcopiesifyouplease.Youmaymodifythebookforyourown

use.Thecatchisthatifyoumakemodiﬁcationsandyoudistributethemodiﬁedversion,

ormakeuseofportionsinexcessoffairuseinanotherwork,thenyoumustalsolicense

thenewworkwiththeGFDL.Sothebookhaslotsofinherentfreedom,andnoone

isallowedtodistributeaderivativeworkthatrestrictsthesefreedoms.(Seethelicense

itselfforalltheexactdetailsoftheadditionalrightsyouhavebeengiven.)

Noticethatinitiallymostpeoplearestruckbythenotionthatthisbookisfree(the

Frenchwouldsaygratis,atnocost).Anditis.However,itismoreimportantthatthe

bookhasfreedom(theFrenchwouldsaylibert´e,liberty).Itwillnevergo“outofprint”

norwillthereeverbetrivialupdatesdesignedonlytofrustratetheusedbookmarket.

Thoseconsideringteachingacoursewiththisbookcanexamineitthoroughlyinadvance.

Addingnewexercisesornewsectionshasbeenpurposelymadeveryeasy,andthehope

isthatotherswillcontributethesemodiﬁcationsbackforincorporationintothebook,

forthebeneﬁtofall.

Dependingonhowyoureceivedyourcopy,youmaywanttocheckforthelatest

version(andothernews)at http://linear.ups.edu/ .

TopicsTheﬁrsthalfofthistext(throughChapterM[219])isbasicallyacoursein

matrixalgebra,thoughthefoundationofsomemoreadvancedideasisalsobeingformed

intheseearlysections.Vectorsarepresentedexclusivelyascolumnvectors(sincewealso

havethetypographicfreedomtoavoidwritingacolumnvectorinlineasthetransposeof

arowvector),andlinearcombinationsarepresentedveryearly.Spans,nullspacesand

columnspacesarealsopresentedearly,simplyassets,savingmostoftheirvectorspace

propertiesforlater,sotheyarefamiliarobjectsbeforebeingscrutinizedcarefully.

Youcannotdoeverythingearly,soinparticularmatrixmultiplicationcomeslater

thanusual.However,withadeﬁnitionbuiltonlinearcombinationsofcolumnvectors,

itshouldseemmorenaturalthantheusualdeﬁnitionusingdotproductsofrowswith

columns.Andthisdelayemphasizesthatlinearalgebraisbuiltuponvectoradditionand

scalarmultiplication.Ofcourse,matrixinversesmustwaitformatrixmultiplication,but

thisdoesnotpreventnonsingularmatricesfromoccurringsooner.Vectorspaceproperties

arehintedatwhenvectorandmatrixoperationsareﬁrstdeﬁned,butthenotionofa

vectorspaceissavedforamoreaxiomatictreatmentlater.Oncebasesanddimension

havebeenexploredinthecontextofvectorspaces,lineartransformationsandtheir

matrixrepresentationsfollow.Thegoalofthebookistogoasfarascanonicalforms

andmatrixdecompositionsintheCore,withlesscentraltopicscollectedinasectionof

Topics.

Linearalgebraisanidealsubjectforthenovicemathematicsstudenttolearnhow

todevelopatopicprecisely,withalltherigormathematicsrequires.Unfortunately,

muchofthisrigorseemstohaveescapedthestandardcalculuscurriculum,soformany

universitystudentsthisistheirﬁrstexposuretocarefuldeﬁnitionsandtheorems,and

theexpectationthattheyfullyunderstandthem,tosaynothingoftheexpectationthat

theybecomeproﬁcientinformulatingtheirownproofs.Wehavetriedtomakethistext

ashelpfulaspossiblewiththistransition.Everydeﬁnitionisstatedcarefully,setapart

fromthetext.Likewise,everytheoremiscarefullystated,andalmosteveryonehasa

completeproof.Theoremsusuallyhavejustoneconclusion,sotheycanbereferenced

preciselylater.Deﬁnitionsandtheoremsarecatalogedinorderoftheirappearancein

thefrontofthebook,andalphabeticalorderintheindexattheback.Alongtheway,

therearediscussionsofsomemoreimportantideasrelatingtoformulatingproofs(Proof

Techniques),whichisadvicemostly.

OriginandHistoryThisbookistheresultoftheconﬂuenceofseveralrelatedevents

andtrends.

•AttheUniversityofPugetSoundweteachaone-semester,post-calculuslinear

algebracoursetostudentsmajoringinmathematics,computerscience,physics,

chemistryandeconomics.BetweenJanuary1986andJune2002,Itaughtthis

courseseventeentimes.FortheSpring2003semester,Ielectedtoconvertmy

coursenotestoanelectronicformsothatitwouldbeeasiertoincorporatethe

inevitableandnearly-constantrevisions.Centraltomynewnoteswasacollection

ofstockexamplesthatwouldbeusedrepeatedlytoillustratenewconcepts.(These

wouldbecometheArchetypes,ChapterA[685].)Itwasonlyashortleaptothen

decidetodistributecopiesofthesenotesandexamplestothestudentsinthetwo

sectionsofthiscourse.Asthesemesterworeon,thenotesbegantolooklesslike

notesandmorelikeatextbook.

•IusedthenotesagainintheFall2003semesterforasinglesectionofthecourse.

Simultaneously,thetextbookIwasusingcameoutinaﬁfthedition.Anewchapter

Version0.70

iii

wasaddedtowardthestartofthebook,andafewadditionalexerciseswereadded

inotherchapters.Thisdemandedtheannoyanceofreworkingmynotesandlist

ofsuggestedexercisestoconformwiththechangednumberingofthechaptersand

exercises.IhadanalmostidenticalexperiencewiththethirdcourseIwasteaching

thatsemester.IalsolearnedthatinthenextacademicyearIwouldbeteaching

acoursewheremytextbookofchoicehadgoneoutofprint.Ifelttherehadto

beabetteralternativetohavingtheorganizationofmycoursesbuetedbythe

economicsoftraditionaltextbookpublishing.

•IhadusedT E XandtheInternetformanyyears,sotherewaslittletostandinthe

wayoftypesetting,distributingand“marketing”afreebook.Withrecreational

andprofessionalinterestsinsoftwaredevelopment,Ihadlongbeenfascinatedbythe

open-sourcesoftwaremovement,asexempliﬁedbythesuccessofGNUandLinux,

thoughpublic-domainT E Xmightalsodeservemention.Obviously,thisbookisan

attempttocarryoverthatmodelofcreativeendeavortotextbookpublishing.

•AsasabbaticalprojectduringtheSpring2004semester,Iembarkedonthecurrent

projectofcreatingafreely-distributablelinearalgebratextbook.(Noticetheim-

pliedﬁnancialsupportoftheUniversityofPugetSoundtothisproject.)Mostof

thematerialwaswrittenfromscratchsincechangesinnotationandapproachmade

muchofmynotesoflittleuse.ByAugust2004Ihadwrittenhalfthematerial

necessaryforourMath232course.TheremaininghalfwaswrittenduringtheFall

2004semesterasItaughtanothertwosectionsofMath232.

•ItaughtasinglesectionofthecourseintheSpring2005semester,whilemycol-

league,ProfessorMartinJackson,graciouslytaughtanothersectionfromthecon-

stantlyshiftingsandsthatwasthisproject(version0.30).Hismanysuggestions

havehelpedimmeasurably.FortheFall2005semester,Itaughttwosectionsofthe

coursefromversion0.50.

However,muchofmymotivationforwritingthisbookiscapturedbythesentiments

expressedbyH.M.CundyandA.P.RolletintheirPrefacetotheFirstEditionofMath-

ematicalModels(1952),especiallytheﬁnalsentence,

Thisbookwasbornintheclassroom,andarosefromthespontaneousinterest

ofaMathematicalSixthintheconstructionofsimplemodels.Adesireto

showthateveninmathematicsonecouldhavefunledtoanexhibitionof

theresultsandattractedconsiderableattentionthroughouttheschool.Since

thentheSherbornecollectionhasgrown,ideashavecomefrommanysources,

andwidespreadinteresthasbeenshown.Itseemsthereforedesirabletogive

permanentformtothelessonsofexperiencesothatotherscanbeneﬁtby

themandbeencouragedtoundertakesimilarwork.

HowToUseThisBookChapters,Theorems,etc.arenotnumberedinthisbook,

butareinsteadreferencedbyacronyms.ThismeansthatTheoremXYZwillalwaysbe

TheoremXYZ,nomatterifnewsectionsareadded,orifanindividualdecidestoremove

Version0.70

Beezer Robert A. - A First Course In Linear Algebra.pdf

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