Linear Algebra: Step by Step

617 Pages·2013·8.43 MB·English

by  Kuldeep Singh

Linear Algebra

Linear Algebra
Step by Step
Kuldeep Singh
SeniorLecturerinMathematics
UniversityofHertfordshire
3
3
GreatClarendonStreet,Oxford,OX26DP,
UnitedKingdom
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Preface
My interest in mathematics began at school. I am originally of Sikh descent, and as a young child
oftenfoundEnglishdifficulttocomprehend,butIdiscoveredanaffinitywithmathematics,auniversal
languagethatIcouldbegintolearnfromthesamestartpointasmypeers.
Linearalgebraisafundamentalareaofmathematics,andisarguablythemostpowerfulmathematical
tooleverdeveloped.Itisacoretopicofstudywithinfieldsasdiverseasbusiness,economics,engineer-
ing,physics,computerscience,ecology,sociology,demographyandgenetics.Foranexampleoflinear
algebraatwork,oneneedlooknofurtherthantheGooglesearchengine,whichreliesonlinearalgebra
toranktheresultsofasearchwithrespecttorelevance.
Mypassionhasalwaysbeentoteach,andIhaveheldthepositionofSeniorLecturerinMathematics
at the University of Hertfordshire for over twenty years, where I teach linear algebra to entry level
undergraduates.IamalsotheauthorofEngineeringMathematicsThroughApplications,abookthatI
amproudtosayisusedwidelyasthebasisforundergraduatestudiesinmanydifferentcountries.Ialso
hostandregularlyupdateawebsitededicatedtomathematics.
AttheUniversityofHertfordshirewehaveoveronehundredmathematicsundergraduates.Inthe
pastwehavebasedourlinearalgebracoursesonvariousexistingtextbooks,butingeneralstudents
havefoundthemhardtodigest;oneofmyprimaryconcernshasbeeninfindingrigorous,yetaccessible
textbookstorecommendtomystudents.Becauseofthepopularityofmypreviouslypublishedbook,I
havefeltcompelledtoconstructabookonlinearalgebrathatbridgestheconsiderabledividebetween
schoolandundergraduatemathematics.
IamsomewhatfortunateinthatIhavehadsomanystudentstoassistmeinevaluatingeachchapter.
Inresponsetotheirreactions,Ihavemodified,expandedandaddedsectionstoensurethatitscontent
entirelyencompassestheabilityofstudentswithalimitedmathematicalbackground,aswellasthe
moreadvancedscholarsundermytutelage.Ibelievethatthishasallowedmetocreateabookthatis
unparalleledinthesimplicityofitsexplanation,yetcomprehensiveinitsapproachtoeventhemost
challengingaspectsofthistopic.
Level
This book is intended for first- and second-year undergraduates arriving with average mathematics
grades. Many students find the transition between school and undergraduate mathematics difficult,
andthisbookspecificallyaddressesthatgapandallowsseamlessprogression.Itassumeslimitedprior
mathematicalknowledge,yetalsocoversdifficultmaterialandanswerstoughquestionsthroughtheuse
ofclearexplanationandawealthofillustrations.Theemphasisofthebookisonstudentslearningfor
themselvesbygraduallyabsorbingclearlypresentedtext,supportedbypatterns,graphsandassociated
questions.Thetextallowsthestudenttograduallydevelopanunderstandingofatopic,withoutthe
needforconstantadditionalsupportfromatutor.
PedagogicalIssues
Thestrengthofthetextisinthelargenumberofexamplesandthestep-by-stepexplanationofeachtopic
asitisintroduced.Itiscompiledinawaythatallowsdistancelearning,withexplicitsolutionstoallof
vi PREFACE
thesetproblemsfreelyavailableonline<http://www.oup.co.uk/companion/singh>.Themiscellaneous
exercisesattheendofeachchaptercomprisequestionsfrompastexampapersfromvariousuniversities,
helpingtoreinforcethereader’sconfidence.Alsoincludedareshorthistoricalbiographiesoftheleading
playersinthefieldoflinearalgebra.Thesearegenerallyplacedatthebeginningofasectiontoengage
theinterestofthestudentfromtheoutset.
Publishedtextbooksonthissubjecttendtoberatherstaticintheirpresentation.Bycontrast,mybook
strivestobesignificantlymoredynamic,andencouragestheengagementofthereaderwithfrequent
questionandanswersections.Thequestion–answerelementissprinkledliberallythroughoutthetext,
consistentlytestingthestudent’sunderstandingofthemethodsintroduced,ratherthanrequiringthem
torememberbyrote.
Thesimpleyetconcisenatureofitscontentisspecificallydesignedtoaidtheweakerstudent,but
itsrigorousapproachandcomprehensivemannermakeitentirelyappropriatereferencematerialfor
mathematiciansateverylevel.IncludedintheonlineresourcewillbeaselectionofMATLABscripts,
providedforthosestudentswhowishtoprocesstheirworkusingacomputer.
Finally,itmustbeacknowledgedthatlinearalgebracanappearabstractwhenfirstencounteredby
astudent.Toshowoffsomeofitspossibilitiesandpotential,interviewswithleadingacademicsand
practitionershavebeenplacedbetweenchapters,givingreadersatasteofwhatmaybetocomeonce
theyhavemasteredthispowerfulmathematicaltool.
Acknowledgements
IwouldparticularlyliketothankTimothyPeacockforhissignficanthelpinimprovingthistext.In
additionIwanttothankSandraStarkeforherconsiderablecontributioninmakingthistextaccessible.
ThankstoototheOUPteam,inparticularKeithMansfield,VikiMortimer,SmitaGuptaandClare
Charles.
Dedication
ToShaheedBibiParamjitKaur
Contents
1 LinearEquationsandMatrices 1
1.1 SystemsofLinearEquations 1
1.2 GaussianElimination 12
1.3 VectorArithmetic 27
1.4 ArithmeticofMatrices 41
1.5 MatrixAlgebra 57
1.6 TheTransposeandInverseofaMatrix 75
1.7 TypesofSolutions 91
1.8 TheInverseMatrixMethod 105
DesHighamInterview 127
2 EuclideanSpace 129
2.1 PropertiesofVectors 129
2.2 FurtherPropertiesofVectors 143
2.3 LinearIndependence 159
2.4 BasisandSpanningSet 171
ChaoYangInterview 190
3 GeneralVectorSpaces 191
3.1 IntroductiontoGeneralVectorSpaces 191
3.2 SubspaceofaVectorSpace 202
3.3 LinearIndependenceandBasis 216
3.4 Dimension 229
3.5 PropertiesofaMatrix 239
3.6 LinearSystemsRevisited 254
JanetDrewInterview 275
4 InnerProductSpaces 277
4.1 IntroductiontoInnerProductSpaces 277
4.2 InequalitiesandOrthogonality 290
4.3 OrthonormalBases 306
4.4 OrthogonalMatrices 321
AnshulGuptaInterview 338
5 LinearTransformations 339
5.1 IntroductiontoLinearTransformations 339
5.2 KernelandRangeofaLinearTransformation 352
5.3 RankandNullity 364
5.4 InverseLinearTransformations 372
5.5 TheMatrixofaLinearTransformation 389
5.6 CompositionandInverseLinearTransformations 407
PetrosDrineasInterview 429
viii CONTENTS
6 DeterminantsandtheInverseMatrix 431
6.1 DeterminantofaMatrix 431
6.2 DeterminantofOtherMatrices 439
6.3 PropertiesofDeterminants 455
6.4 LUFactorization 472
FrançoiseTisseurInterview 490
7 EigenvaluesandEigenvectors 491
7.1 IntroductiontoEigenvaluesandEigenvectors 491
7.2 PropertiesofEigenvaluesandEigenvectors 503
7.3 Diagonalization 518
7.4 DiagonalizationofSymmetricMatrices 533
7.5 SingularValueDecomposition 547
BriefSolutions 567
Index 605
.............................................................................................
1
Linear Equations
and Matrices
.............................................................................................
SECTION 1.1 SystemsofLinearEquations
Bytheendofthissectionyouwillbeableto
● solvealinearsystemofequations
● plotlineargraphsanddeterminethetypeofsolutions
1.1.1 Introductiontolinearalgebra
Weareallfamiliarwithsimpleone-lineequations.Anequationiswheretwomathematical
expressionsaredefinedasbeingequal.Given3x=6,wecanalmostintuitivelyseethatx
mustequal2.
However,thesolutionisn’talwaysthiseasytofind,andthefollowingexampledemon-
strateshowwecanextractinformationembeddedinmorethanonelineofinformation.
ImagineforamomentthatJohnhasboughttwoicecreamsandtwodrinksfor£3.00.
HowmuchdidJohnpayforeachitem?
Letx=costoficecreamandy=costofdrink,thentheproblemcanbewrittenas
2x+2y=3
Atthispoint,itisimpossibletofindauniquevalueforthecostofeachitem.However,youare
thentoldthatJaneboughttwoicecreamsandonedrinkfor£2.50.Withthisadditionalinforma-
tion,wecanmodeltheproblemasasystemofequationsandlookforuniquevaluesforthecost
oficecreamsanddrinks.Theproblemcannowbewrittenas
2x+2y = 3
2x+y = 2.5
Usingabitofguesswork,wecanseethattheonlysensiblevaluesforxandythatsatisfyboth
equationsarex=1andy=0.5.Thereforeanicecreammusthavecost£1.00andadrink£0.50.
Ofcourse,thisisanextremelysimpleexample,thesolutiontowhichcanbefoundwith
aminimumofcalculation,butlargersystemsofequationsoccurinareaslikeengineering,
scienceandfinance.Inordertoreliablyextractinformationfrommultiplelinearequations,
we need linear algebra. Generally, the complex scientific, or engineering problem can be
solvedbyusinglinearalgebraonlinearequations.