Vectors, Pure and Applied: A General Introduction to Linear Algebra

457 Pages·2013·1.55 MB·English

more information - www.cambridge.org/9781107033566
VECTORS, PURE AND APPLIED
AGeneralIntroductiontoLinearAlgebra
Many books on linear algebra focus purely on getting students through exams, but this
text explains both the how and the why of linear algebra and enables students to begin
thinking like mathematicians. The author demonstrates how different topics (geometry,
abstract algebra, numerical analysis, physics) make use of vectors in different ways, and
howthesewaysareconnected,preparingstudentsforfurtherworkintheseareas.
The book is packed with hundreds of exercises ranging from the routine to the
challenging.Sketchsolutionsoftheeasierexercisesareavailableonline.
t. w. ko¨rner isProfessorofFourierAnalysisintheDepartmentofPureMathematics
and Mathematical Statistics at the University of Cambridge. His previous books include
FourierAnalysisandThePleasuresofCounting.

VECTORS, PURE AND APPLIED
A General Introduction to Linear Algebra
T. W. KO¨ RNER
TrinityHall,Cambridge
cambridge university press
Cambridge,NewYork,Melbourne,Madrid,CapeTown,
Singapore,Sa˜oPaulo,Delhi,MexicoCity
CambridgeUniversityPress
TheEdinburghBuilding,CambridgeCB28RU,UK
PublishedintheUnitedStatesofAmericabyCambridgeUniversityPress,NewYork
www.cambridge.org
Informationonthistitle:www.cambridge.org/9781107033566
(cid:2)C T.W.Ko¨rner2013
Thispublicationisincopyright.Subjecttostatutoryexception
andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2013
PrintedandboundintheUnitedKingdombytheMPGBooksGroup
AcataloguerecordforthispublicationisavailablefromtheBritishLibrary
LibraryofCongressCataloguinginPublicationdata
Ko¨rner,T.W.(ThomasWilliam),1946–
Vectors,pureandapplied:ageneralintroductiontolinearalgebra/T.W.Ko¨rner.
pages cm
Includesbibliographicalreferencesandindex.
ISBN978-1-107-03356-6(hardback)–ISBN978-1-107-67522-3(paperback)
1.Vectoralgebra. 2.Algebras,Linear. I.Title.
QA200.K67 2013
516(cid:3).182–dc23 2012036797
ISBN978-1-107-03356-6Hardback
ISBN978-1-107-67522-3Paperback
Additionalresourcesforthispublicationatwww.dpmms.cam.ac.uk/∼twk/
CambridgeUniversityPresshasnoresponsibilityforthepersistenceor
accuracyofURLsforexternalorthird-partyinternetwebsitesreferredto
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Ingeneralthepositionasregardsallsuchnewcalculiisthis.–Thatonecannotaccomplishbythem
anythingthatcouldnotbeaccomplishedwithoutthem.However,theadvantageis,that,providedthat
suchacalculuscorrespondstotheinmostnatureoffrequentneeds,anyonewhomastersitthoroughly
isable–withouttheunconsciousinspirationwhichnoonecancommand–tosolvetheassociated
problems,eventosolvethemmechanicallyincomplicatedcasesinwhich,withoutsuchaid,even
geniusbecomespowerless....Suchconceptionsunite,asitwere,intoanorganicwhole,countless
problemswhichotherwisewouldremainisolatedandrequirefortheirseparatesolutionmoreorless
ofinventivegenius.
(GaussWerke,Bd.8,p.298(quotedbyMoritz[24]))
For many purposes of physical reasoning, as distinguished from calculation, it is desirable to
avoid explicitly introducing ...Cartesian coordinates, and to fix the mind at once on a point of
spaceinsteadofitsthreecoordinates,andonthemagnitudeanddirectionofaforceinsteadofits
threecomponents....Iamconvincedthattheintroductionoftheidea[ofvectors]willbeofgreat
usetousinthestudyofallpartsofoursubject,andespeciallyinelectrodynamicswherewehaveto
dealwithanumberofphysicalquantities,therelationsofwhichtoeachothercanbeexpressedmuch
moresimplyby[vectorialequationsrather]thanbytheordinaryequations.
(MaxwellATreatiseonElectricityandMagnetism[21])
We [Halmos and Kaplansky] share a love of linear algebra. ...And we share a philosophy about
linearalgebra:wethinkbasis-free,wewritebasis-free,butwhenthechipsaredownweclosethe
officedoorandcomputewithmatriceslikefury.
(KaplanskyinPaulHalmos:CelebratingFiftyYearsofMathematics[17])
MarcoPolodescribesabridge,stonebystone.
‘Butwhichisthestonethatsupportsthebridge?’KublaiKhanasks.
‘Thebridgeisnotsupportedbyonestoneoranother,’Marcoanswers,‘butbythelineofthearch
thattheyform.’
KublaiKhanremainssilent,reflecting.Thenheadds:‘Whydoyouspeaktomeofthestones?It
isonlythearchthatmatterstome.’
Poloanswers:‘Withoutstonesthereisnoarch.’
(CalvinoInvisibleCities(translatedbyWilliamWeaver)[8])

Contents
Introduction pagexi
PARTI FAMILIARVECTORSPACES 1
1 Gaussianelimination 3
1.1 Twohundredyearsofalgebra 3
1.2 Computationalmatters 8
1.3 Detachedcoefficients 12
1.4 Anotherfiftyyears 15
1.5 Furtherexercises 18
2 Alittlegeometry 20
2.1 Geometricvectors 20
2.2 Higherdimensions 24
2.3 Euclideandistance 27
2.4 Geometry,planeandsolid 32
2.5 Furtherexercises 36
3 Thealgebraofsquarematrices 42
3.1 Thesummationconvention 42
3.2 Multiplyingmatrices 43
3.3 Morealgebraforsquarematrices 45
3.4 Decompositionintoelementarymatrices 49
3.5 Calculatingtheinverse 54
3.6 Furtherexercises 56
4 Thesecretlifeofdeterminants 60
4.1 Theareaofaparallelogram 60
4.2 Rescaling 64
4.3 3×3determinants 66
4.4 Determinantsofn×nmatrices 72
4.5 Calculatingdeterminants 75
4.6 Furtherexercises 81
vii
viii Contents
5 Abstractvectorspaces 87
5.1 ThespaceCn 87
5.2 Abstractvectorspaces 88
5.3 Linearmaps 91
5.4 Dimension 95
5.5 Imageandkernel 103
5.6 Secretsharing 111
5.7 Furtherexercises 114
6 LinearmapsfromFntoitself 118
6.1 Linearmaps,basesandmatrices 118
6.2 Eigenvectorsandeigenvalues 122
6.3 Diagonalisationandeigenvectors 125
6.4 LinearmapsfromC2toitself 127
6.5 Diagonalisingsquarematrices 132
6.6 Iteration’sartfulaid 136
6.7 LU factorisation 141
6.8 Furtherexercises 146
7 Distancepreservinglinearmaps 160
7.1 Orthonormalbases 160
7.2 Orthogonalmapsandmatrices 164
7.3 RotationsandreflectionsinR2andR3 169
7.4 ReflectionsinRn 174
7.5 QRfactorisation 177
7.6 Furtherexercises 182
8 Diagonalisationfororthonormalbases 192
8.1 Symmetricmaps 192
8.2 Eigenvectorsforsymmetriclinearmaps 195
8.3 Stationarypoints 201
8.4 Complexinnerproduct 203
8.5 Furtherexercises 207
9 Cartesiantensors 211
9.1 Physicalvectors 211
9.2 GeneralCartesiantensors 214
9.3 Moreexamples 216
9.4 Thevectorproduct 220
9.5 Furtherexercises 227
Contents ix
10 Moreontensors 233
10.1 Sometensorialtheorems 233
10.2 A(very)littlemechanics 237
10.3 Left-hand,right-hand 242
10.4 Generaltensors 244
10.5 Furtherexercises 247
PARTII GENERALVECTORSPACES 257
11 Spacesoflinearmaps 259
11.1 AlookatL(U,V) 259
11.2 AlookatL(U,U) 266
11.3 Duals(almost)withoutusingbases 269
11.4 Dualsusingbases 276
11.5 Furtherexercises 283
12 PolynomialsinL(U,U) 291
12.1 Directsums 291
12.2 TheCayley–Hamiltontheorem 296
12.3 Minimalpolynomials 301
12.4 TheJordannormalform 307
12.5 Applications 312
12.6 Furtherexercises 316
13 Vectorspaceswithoutdistances 329
13.1 Alittlephilosophy 329
13.2 Vectorspacesoverfields 329
13.3 Errorcorrectingcodes 334
13.4 Furtherexercises 340
14 Vectorspaceswithdistances 344
14.1 Orthogonalpolynomials 344
14.2 Innerproductsanddualspaces 353
14.3 Complexinnerproductspaces 359
14.4 Furtherexercises 364
15 Moredistances 369
15.1 DistanceonL(U,U) 369
15.2 Innerproductsandtriangularisation 376
15.3 Thespectralradius 379
15.4 Normalmaps 383
15.5 Furtherexercises 387