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Saikia-LinearAlgebra book1 February18,2014 14:0 Linear Algebra i Saikia-LinearAlgebra book1 February18,2014 14:0 Linear Algebra Second Edition Promode Kumar Saikia North-EasternHillUniversity iii Saikia-LinearAlgebra book1 February18,2014 14:0 No part of this eBook may be used or reproduced in any manner whatsoever without the publisher’s prior written consent. Copyright © 2014 Pearson India Education Services Pvt. Ltd This eBook may or may not include all assets that were part of the print version. The publisher reserves the right to remove any material in this eBook at any time. ISBN: 9789332522145 eISBN: 9789332540521 Head Office: 7th Floor, Knowledge Boulevard, A-8(A) Sector 62, Noida 201 309, India. Registered Office: Module G4, Ground Floor, Elnet Software City, TS-140, Block 2 & 9, Rajiv Gandhi Salai, Taramani, Chennai, Tamil Nadu 600113, Fax : 080-30461003, Phone: 080-30461060, www.pearson.co.in, Email id: [email protected] iv Saikia-LinearAlgebra book1 February18,2014 14:0 Contents Preface ix Preface to the Second Edition xiii A Note to Students xv List of Symbols xvii 1 Matrices 1 1.1 Introduction 1 1.2 BasicConcepts 1 1.3 MatrixOperationsandTheirProperties 15 1.4 InvertibleMatrices 27 1.5 TransposeofaMatrix 32 1.6 PartitionofMatrices;BlockMultiplication 36 1.7 GroupsandFields 45 2 Systems of Linear Equations 49 2.1 Introduction 49 2.2 GaussianElimination 49 2.3 ElementaryRowOperations 55 2.4 RowReduction 65 2.5 InvertibleMatricesAgain 77 2.6 LUFactorization 82 2.7 Determinant 96 v Saikia-LinearAlgebra book1 February18,2014 14:0 vi Contents 3 Vector Spaces 114 3.1 Introduction 114 3.2 BasicConcepts 115 3.3 LinearIndependence 127 3.4 BasisandDimension 135 3.5 SubspacesAgain 147 3.6 RankofaMatrix 153 3.7 OrthogonalityinRn 163 3.8 BasesofSubspaces 178 3.9 QuotientSpace 184 4 Linear Maps and Matrices 191 4.1 Introduction 191 4.2 BasicConcepts 191 4.3 AlgebraofLinearMaps 204 4.4 Isomorphism 215 4.5 MatricesofLinearMaps 221 5 Linear Operators 237 5.1 Introduction 237 5.2 PolynomialsOverFields 238 5.3 CharacteristicPolynomialsandEigenvalues 243 5.4 MinimalPolynomial 271 5.5 InvariantSubspaces 283 5.6 SomeBasicResults 298 5.7 RealQuadraticForms 310 6 Canonical Forms 321 6.1 Introduction 321 6.2 PrimaryDecompositionTheorem 321 6.3 JordanForms 329 Saikia-LinearAlgebra book1 February18,2014 14:0 Contents vii 7 Bilinear Forms 346 7.1 Introduction 346 7.2 BasicConcepts 346 7.3 LinearFunctionalsandDualSpace 355 7.4 SymmetricBilinearForms 360 7.5 GroupsPreservingBilinearForms 374 8 Inner Product Spaces 380 8.1 Introduction 380 8.2 HermitianForms 380 8.3 InnerProductSpace 385 8.4 Gram–SchmidtOrthogonalizationProcess 390 8.5 Adjoints 403 8.6 UnitaryandOrthogonalOperators 409 8.7 NormalOperators 416 Bibliography 430 Index 431 Saikia-LinearAlgebra book1 February18,2014 14:0 Saikia-LinearAlgebra book1 February18,2014 14:0 Preface Thisbookistheoutcomeofagrowingrealization,sharedbymycolleagues,thatthereisaneedfor acomprehensivetextbookinlinearalgebrawhosemainemphasisshouldbeonclarityofexposition. Thereareseveralexcellenttextbooksavailablecurrently;however,theperceptionisthateachofthese hasitsownareaofexcellenceleavingroomforimprovement.Thisperceptionhasguidedtheapproach tosometopicsofthisbook.Forthecontentsofthebook,Ihavedrawnonmyexperienceofteaching a full semester course in linear algebra overthe years for postgraduateclasses in the North-Eastern HillUniversityinShillong,India.Theinputsfromsomecolleaguesfromundergraduatecollegeshave alsohelped. My main concern has always been with simplicity and clarity, and an effort has been made to avoidcumbersomenotations.Ihaveoptedforinformaldiscussionsinsteadofgivingdefinitionswhich appearcluttered-up.Overall,ouraimhasbeentohelpreadersacquireafeelingforthesubject.Plenty ofexamplesandnumerousexercisesarealsoincludedinthisbook. Chapter1 introducesmatricesandmatrixoperationsandexploresthe algebraicstructuresofsets ofmatriceswhileemphasizingthesimilaritieswithmorefamiliarstructures.Theroleofunitmatrices in the ringstructureofmatricesisalso discussed.Block operationsofpartitionedmatricesare quite usefulinlaterchapters.Thischapterdiscussessuchmatricestomakereaderscomfortablewiththeir uses. Chapter 2 comprehensively covers the treatment of solutions of systems of linear equations by row reduction with the help of elementary row/column operations. Elementary matrices appear naturally;theirusefulnessinanalysingmatrices,especiallyinvertiblematrices,isalsoexamined,and a section on propertiesof determinantsis also included in this chapter. Determinantsare defined in termsofexpansionsbyminorsalongthefirstrow;bydoingso,ithasbecomepossibletogiveproofs ofpropertiesofdeterminantsofarbitraryordersaccessibletoevenundergraduatestudents.Itshould be noted that these propertiesare well-knownand used frequentlybut hardly provedin classrooms. Chapter 3 begins by introducing the basic concepts related to vector spaces. Ample examples are providedforconceptslikelinearindependence,basisandcoordinatestomakeiteasierforanaverage student.Awholesectionofthischapterisdevotedtotheideaoftherankofamatrixincomputations as well as in theory. Rank of a matrix is defined through the row space and the column space of the matrix;thisapproachhasthe advantageof workingwith ideaslike linearindependenceto make relevantproofsmoretransparent.Computationsofbasesofsumsandintersectionsofsubspaceshave always been difficult for students and an attempt has been made to remove the difficulties of such computations. The easy-pacedtreatment of the topics of these three chapters makes this part of the booksuitableforbothstudentsandteachersofundergraduatecourses. Chapters4 to8 dealadequatelywith theessentials inlinearalgebrafora postgraduatestudentin mathematics. More practically, the topics cover the requirementsof the NET syllabus. A brief look atthe contentsofthese chaptersfollows.Linearmapsbetweenvectorspacesarestudiedin detailin Chapter4.Theinterplaybetweenlinearmapsandmatricesisstressedthroughoutthischapter.Other ix Saikia-LinearAlgebra book1 February18,2014 14:0 x Preface important concepts, such as isomorphism, dimension formula and similarity, are dealt with in this chapter. Projections as well as nilpotent maps and matrices are also introduced so that readers are familiarwiththemlongbeforetheiractualapplications.Chapter5isalongone;thegoalistoobtain thediagonalizationtheorems.However,themainemphasisistocarefullydeveloptheconcepts,such as eigenvalues,characteristic polynomials,minimal polynomialsand invariantsubspaces, which are essentialinmanybranchesofhighermathematics.Cyclicsubspacesandcompanionmatricesarealso treatedhere.Chapter6isdevotedtocanonicalformsofmatrices.Ashorterandmoreaccessibletreat- mentofJordanformisprovided.Primarydecompositiontheoremandrationalcanonicalformsarethe othertwotopicsinthischapter.Chapter7discussesbilinearforms.Amethodfordiagonalizingsym- metricmatricesaswellasquadraticformsisgivenhere.Sylvester’sclassicalresultforrealsymmetric matricesisalsoincluded.Chapter8dealswithcertainessentialconceptswhichcanbetreatedinthe frameworkof innerproductspacesand are introducedthroughhermitianforms.The mainobjective ofthischapteristo obtainthe orthogonaldiagonalizationofhermitianandrealsymmetricmatrices. Standardtopics,suchasGram-Schmidtprocess,adjoints,self-adjointandnormaloperators,arethor- oughlyexaminedinthischapterleadingtotheSpectraltheorem.Unitaryandorthogonaloperatorsare theotherkeytopicsofthischapter. The final chapter, Chapter 9, is devoted to a few topics which are must for a student of linear algebra but unfortunately do not find a place in the syllabi of linear algebra in most of the Indian universities.Thechapterbeginswithadiscussionofrigidmotionsandthecanonicalformsfororthog- onaloperators.Manyapplicationsoflinearalgebraindiversedisciplinesdependonthetheoryofreal quadraticformsandrealsymmetricmatrices;asexamplesofsuchapplications,thischapterdiscusses the classifications of conics and quadrics as well as the problems of constrained optimization, and relativeextremaofreal-valuedfunctions.Tofacilitatethediscussionoftheseproblems,positivedef- initematricesarealsointroduced.Singularvaluedecompositionsofrealorcomplexmatricesreveal importantpropertiesofsuchmatricesandleadtoamazingapplications.Thelastsectionofthechapter dealswithsingularvaluedecompositions;asanapplication,Moore–Penroseinversesofmatricesare brieflydiscussed. Numerousexercisesareprovidedforalmostallthesectionsofthebook.Theseexercisesforman integralpartof the text; attempts to solve these will enhance the understandingof the material they dealwith.Awordaboutthetrue/falsequestionsincludedinthisbook:We,attheNorth-EasternHill University,havebeenencouragingthepracticeofincludingsuchtrue/falsequestionsinexamination papers.Wehopethattheinclusionofsuchquestionsinthisbookwillhelpspreadthepracticetoother mathematicsdepartmentsofthecountry. Mythoughtsaboutthesubjectmatterofthisbookhavebeenshapedbyvariousbooksandarticles onalgebraandlinearalgebrabymasterexpositorssuchasHalmos,Herstein,Artinandothers.Their influence on this bookis undeniable.I take this opportunityto acknowledgemy indebtednessto all ofthem.Ihavealso beengreatlybenefitedbythe textbookslisted in thebibliography;I expressmy gratitudetoalltheauthorsofthesetextbooks.Thematerialaboutisometryinthelastchapterclosely followsKumaresan’slovelyarticleonisometrieswhichappearedintheMathematicsNewsletter,vol. 14,March2005. Above all, my colleagues in the Mathematics Department of the North-Eastern Hills University deservespecialthanksforhelpingmeinsomanywaysduringthepreparationofthisbook.Professor M.B. Rege and Professor H. Mukherjee were always ready with suggestions for me; their encour- agementkeptmegoing.Innumerablediscussionswithmyyoungercolleagues,AshishDas,A.Tiken Singh,A.M.Buhphang,S.Dutta,J.SinghandDeepakSubedi,helpedmeimmenselytogivethefinal shapetothemanuscript,especiallyinpreparingvariousexercises.A.TikenSinghandAshishDasalso
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