The linear algebra a beginning graduate student ought to know

510 Pages·2012·4.49 MB·English

by  Jonathan S. Golan (auth.)

The Linear Algebra a Beginning Graduate
Student Ought to Know

Jonathan S. Golan
The Linear Algebra
a Beginning Graduate
Student Ought
to Know
Third Edition
Prof.JonathanS.Golan
Dept.ofMathematics
UniversityofHaifa
Haifa
Israel
ISBN978-94-007-2635-2 e-ISBN978-94-007-2636-9
DOI10.1007/978-94-007-2636-9
SpringerDordrechtHeidelbergLondonNewYork
LibraryofCongressControlNumber:2012933373
MathematicsSubjectClassification(2010): 15-XX,16-XX,17-XX,65F30
©SpringerScience+BusinessMediaB.V.2004,2007,2012
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To thememoryofHemda,mywifeofover40
years:
Andtomygrandchildren: Shachar, Eitan,
Sarel, Nachshon, Yarden, Itamar,Roni,and
Naomi

For Whom Is This Book Written?
Crow’sLaw:Donotthinkwhatyouwanttothinkuntilyouknowwhatyou
oughttoknow.1
Linearalgebraisa living,activebranchof mathematicalresearch whichis central
toalmostallotherareasofmathematicsandwhichhasimportantapplicationsinall
branchesofthephysicalandsocialsciencesandinengineering.However,inrecent
years the content of linear algebra courses required to complete an undergraduate
degreeinmathematics—andevenmoresoinotherareas—atallbutthemostded-
icated universities, has been depleted to the extent that it falls far short of what is
in fact needed for graduate study and research or for real-world application. This
is true not only in the areas of theoretical work but also in the areas of computa-
tionalmatrixtheory,whicharebecomingmoreandmoreimportanttotheworking
researcher as personal computers become a common and powerful tool. Students
arenotonlylessabletoformulateorevenfollowmathematicalproofs,theyarealso
lessabletounderstandtheunderlyingmathematicsofthenumericalalgorithmsthey
mustuse.Theresultingknowledgegaphasledtofrustrationandrecriminationon
the part of both students and faculty alike, with each silently—and sometimes not
sosilently—blamingtheotherfortheresultingstateofaffairs.Thisbookiswritten
with the intention of bridging that gap. It was designed be used in one or more of
severalpossibleways:
(1) Asaself-studyguide;
(2) Asatextbookforacourseinadvancedlinearalgebra,eitherattheupper-class
undergraduateleveloratthefirst-yeargraduatelevel;or
(3) Asareferencebook.
It is also designed to be used to prepare for the linear algebra portion of prelim
examsorPh.D.qualifyingexams.
This volume is self-contained to the extent that it does not assume any previ-
ousknowledgeofformallinearalgebra,thoughthereaderisassumedtohavebeen
exposed,atleastinformally,tosomebasicideasortechniques,suchasmatrixma-
nipulationandthesolutionofasmallsystemoflinearequations.Itdoes,however,
1Thislaw,attributedtoJohnCrowofKing’sCollege,London,isquotedbyR.V.Jonesinhisbook
MostSecretWar,Wordsworth,1998(ISBN978-1853266997).
vii
viii ForWhomIsThisBookWritten?
assumeaseriousnessofpurpose,considerablemotivation,andmodicumofmathe-
maticalsophisticationonthepartofthereader.
Thetheoreticalconstructionspresentedhereareillustratedwithalargenumberof
examplestakenfromvariousareasofpureandappliedmathematics.Asinanyarea
ofmathematics,theoryandconcreteexamplesmustgohandinhandandneedtobe
studied together. As the German philosopher Immanuel Kant famously remarked,
conceptswithoutpreceptsareempty,whereaspreceptswithoutconceptsareblind.
The book also contains a large number of exercises, many of which are quite
challenging,whichIhavecomeacrossorthoughtupinoverthirtyyearsofteaching.
Many of these exercises have appeared in print before, in such journals as Ameri-
can Mathematical Monthly, College Mathematics Journal, Mathematical Gazette,
orMathematicsMagazine,invariousmathematicscompetitionsorcirculatedprob-
lem collections, or even on the internet. Some were donated to me by colleagues
andevenstudents,andsomeoriginatedinfilesofoldexamsatvariousuniversities
whichIhavevisitedinthecourseofmycareer.Since,overtheyears,Ididnotkeep
trackoftheirsources,allIcandoisofferacollectiveacknowledgementtoallthose
towhomitisdue.Goodproblemformulators,liketheGodoftheabbotofCiteaux,
knowtheirown.Deliberately,difficultexercisesarenotmarkedwithanasteriskor
othersymbol.Solvingexercisesisanintegralpartoflearningmathematicsandthe
reader is definitely expected to do so, especially when the book is used for self-
study. Try them all and remember the “grook” penned by the Danish genius Piet
Hein:Problemsworthyofattack/Provetheirworthbyhittingback.
Solving a problem using theoretical mathematics is often very different from
solvingitcomputationally,andsostrongemphasisisplacedontheinterplayofthe-
oretical and computational results. Real-life implementation of theoretical results
isperpetuallyplaguedbyerrors: errors inmodeling,errors indataacquisitionand
recording,and errors in thecomputationalprocess itself due toroundoff andtrun-
cation. There are further constraints imposed by limitations in time and memory
availableforcomputation.Thusthemosteleganttheoreticalsolutiontoaproblem
maynotleadtothemostefficientorusefulmethodofsolutioninpractice.Whileno
referenceismadetoparticularcomputersoftware,theconcurrentuseofapersonal
computerequippedsymbolic-manipulationsoftwaresuchasMAPLE,MATHEMAT-
ICA,MATLAB,orMUPADisdefinitelyadvised.
In order to show the “human face” of mathematics, the book also includes a
largenumberofthumbnailphotographsofresearcherswhohavecontributedtothe
developmentofthematerialpresentedinthisvolume.
Acknowledgements Most of the first edition this book was written while I was
avisitorattheUniversityofIowainIowaCityandattheUniversityofCalifornia
inBerkeley.Iwouldliketothankbothinstitutionsforprovidingthefacilitiesand,
more importantly, the mathematical atmosphere which allowed me to concentrate
on writing. Subsequent, extensively revised editions, were prepared after I retired
fromteachingattheUniversityofHaifainApril,2004.
Ihavetalkedtomanystudentsandfacultymembersaboutmyplansforthisbook
and have obtained valuable insights from them. In particular, I would like to ac-
knowledgetheaidofthefollowingcolleaguesandstudentswhowerekindenough
ForWhomIsThisBookWritten? ix
toreadthepreliminaryversionsofthisbookandoffertheircommentsandcorrec-
tions: Prof. Daniel Anderson (University of Iowa), Prof. Adi Ben-Israel (Rutgers
University),Prof.RobertCacioppo(TrumanStateUniversity),Prof.JosephFelsen-
stein (University of Washington), Prof. Ryan Skip Garibaldi (Emory University),
Mr.GeorgeKirkup(UniversityofCalifornia,Berkeley),Dr.DenisSevee(JohnAl-
bert College), Prof. Earl Taft (Rutgers University), Mr. Gil Vernik (University of
Haifa).
Haifa,Israel JonathanS.Golan