Linear Algebra: Concepts and Methods

532 Pages·2012·3.55 MB·English

LinearAlgebra:ConceptsandMethods
Any student of linear algebra will welcome this textbook, which provides a
thorough treatment of this key topic. Blending practice and theory, the book
enablesthereadertolearnandcomprehendthestandardmethods,withanempha-
sisonunderstandinghowtheyactuallywork.Ateverystage,theauthorsarecareful
to ensure that the discussion is no more complicated or abstract than it needs to
be,andfocusesonthemostfundamentaltopics.

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Hundredsofexamplesandexercises,includingsolutions,givestudentsplenty
ofhands-onpractice
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End-of-chaptersectionssummarisematerialtohelpstudentsconsolidatetheir
learning
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Idealasacoursetextandforself-study
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Instructorscanusethemanyexamplesandexercisestosupplementtheirown
assignments
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Both authors have extensive experience of undergraduate teaching and of
preparationofdistancelearningmaterials.
MartinAnthonyisProfessorofMathematicsattheLondonSchoolofEconomics
(LSE),andAcademicCoordinatorforMathematicsontheUniversityofLondon
International Programmes for which LSE has academic oversight. He has over
20 years’ experience of teaching students at all levels of university, and is the
authoroffourbooks,including(withN.L.Biggs)thetextbookMathematicsfor
Economics and Finance: Methods and Modelling (Cambridge University Press,
1996).Healsohasextensiveexperienceofpreparingdistancelearningmaterials.
MicheleHarveylecturesattheLondonSchoolofEconomics,whereshehastaught
anddevelopedthelinearalgebrapartofthecorefoundationcourseinmathematics
forover20years.Herdedicationtohelpingstudentslearnmathematicshasbeen
widelyrecognised.SheisalsoChiefExaminerfortheAdvancedLinearAlgebra
courseontheUniversityofLondonInternationalProgrammesandhasco-authored
withMartin Anthony the study guides for Advanced Linear Algebra and Linear
Algebraonthisprogramme.

Linear Algebra:
Concepts and Methods
MARTIN ANTHONYandMICH ELE HARVEY
DepartmentofMathematics
TheLondonSchoolofEconomicsandPoliticalScience
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/9780521279482
(cid:2)C CambridgeUniversityPress2012
Thispublicationisincopyright.Subjecttostatutoryexception

andtotheprovisionsofrelevantcollectivelicensingagreements,
noreproductionofanypartmaytakeplacewithoutthewritten
permissionofCambridgeUniversityPress.
Firstpublished2012
PrintedintheUnitedKingdomattheUniversityPress,Cambridge
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ISBN 978-0-521-27948-2Paperback
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ToColleen,Alistair,andmyparents.And,justforAlistair,
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here’soneofthosesidewaysmoustaches:
(MA)
ToBill,forhissupportthroughout,andtomyfather,forhis
encouragementtostudymathematics.
(MH)

Contents
Preface pagexiii
Preliminaries:beforewebegin 1
Setsandsetnotation 1
Numbers 2
Mathematicalterminology 3
Basicalgebra 4
Trigonometry 7
Alittlebitoflogic 8
1 Matricesandvectors 10
1.1 Whatisamatrix? 10
1.2 Matrixadditionandscalarmultiplication 11
1.3 Matrixmultiplication 12
1.4 Matrixalgebra 14
1.5 Matrixinverses 16
1.6 Powersofamatrix 20
1.7 Thetransposeandsymmetricmatrices 20
1.8 VectorsinRn 23
1.9 Developinggeometricinsight 27
1.10 Lines 33
1.11 PlanesinR3 39
1.12 LinesandhyperplanesinRn 46
1.13 Learningoutcomes 47
1.14 Commentsonactivities 48
1.15 Exercises 53
1.16 Problems 55
viii Contents
2 Systemsoflinearequations 59
2.1 Systemsoflinearequations 59
2.2 Rowoperations 62
2.3 Gaussianelimination 64
2.4 Homogeneoussystemsandnullspace 75
2.5 Learningoutcomes 81
2.6 Commentsonactivities 82
2.7 Exercises 84
2.8 Problems 86
3 Matrixinversionanddeterminants 90
3.1 Matrixinverseusingrowoperations 90

3.2 Determinants 98
3.3 Resultsondeterminants 104
3.4 Matrixinverseusingcofactors 113
3.5 Leontiefinput–outputanalysis 119
3.6 Learningoutcomes 121
3.7 Commentsonactivities 122
3.8 Exercises 125
3.9 Problems 128
4 Rank,rangeandlinearequations 131
4.1 Therankofamatrix 131
4.2 Rankandsystemsoflinearequations 133
4.3 Range 139
4.4 Learningoutcomes 142
4.5 Commentsonactivities 142
4.6 Exercises 144
4.7 Problems 146
5 Vectorspaces 149
5.1 Vectorspaces 149
5.2 Subspaces 154
5.3 Linearspan 160
5.4 Learningoutcomes 164
5.5 Commentsonactivities 164