Linear Algebra

393 Pages·2017·4.67 MB·English

by  Belkacem Said-Houari (auth.)

Compact Textbooks in Mathematics
Belkacem Said-Houari
Linear
Algebra
CompactTextbooksinMathematics
CompactTextbooksinMathematics
This textbook series presents concise introductions to current topics in
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(cid:2) compact:smallbookspresentingtherelevantknowledge
(cid:2) learningmadeeasy:examplesandexercisesillustratetheapplication
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(cid:2) usefulforlecturers:eachtitlecanserveasbasisandguidelinefora
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Moreinformationaboutthisseriesathttp://www.springer.com/series/11225
BelkacemSaid-Houari
Linear Algebra
BelkacemSaid-Houari
DepartmentofMathematics,CollegeofSciences
UniversityofSharjah
Sharjah,UnitedArabEmirates
ISSN2296-4568 ISSN2296-455X (electronic)
CompactTextbooksinMathematics
ISBN978-3-319-63792-1 ISBN978-3-319-63793-8 (eBook)
DOI10.1007/978-3-319-63793-8
LibraryofCongressControlNumber:2017951442
MathematicsSubjectClassification(2010):15A03,15A04,15A18,15A42,15A63,15B10,11C20,
11E16
©SpringerInternationalPublishingAG2017
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To mydaughterNOUR
vii
Preface
Linear algebra is the study of the algebraic properties of linear transfor-
mations and matrices and it is an essential part of virtually all areas of
mathematics.Itisalsoafundamentalandanextremelypowerfultoolinevery
singledisciplineofsciencesandengineering.
This is a self-contained textbook on linear algebra written in an easy
way, so that it can be accessible to many readers. It begins in (cid:2) Chap. 1
with the simplest linear equation and generalizes many notions about this
equationtosystemsoflinearequations,andthenintroducesthemainideas
usingmatricesandtheirproperties.Webelievethatthisistherightapproach,
since moststudents take the first courseof linear algebra already knowing
something about linear equations, lines, and systems of linear equations.
Then follows a detailed chapter ((cid:2) Chap. 2) on determinants and their
properties where we also study the relationship between determinants and
the inverses of matrices and the use of determinants in solving systems of
linearequations.Weintroducethemainideaswithdetailedproofs.Wealso
investigatesomeparticulardeterminantsthatareveryusefulinapplications.
Inaddition,weexplaininasimplewaywheretheideasofdeterminantscome
fromandhowtheyfittogetherinthewholetheory.
In(cid:2)Chap.3,weintroducetheEuclideanspacesusingverysimplegeo-
metric ideas and then discuss various important inequalities and identities.
These ideas are presentin the theory of general Hilbert spaces in a course
of functional analysis, so it is much better for students to learn them and
understandthemclearlyinEuclideanspaces.
Thecoreof(cid:2)Chap.4isadetaileddiscussionofgeneralvectorspaces
where rigorous proofs to all the main results in this book are given. This
is followed by a chapter ((cid:2) Chap. 5) on linear transformations and their
properties.
In (cid:2)Chap.6,weintroducenotionsconcerningmatricesthroughlinear
transformations, trying to bridge thegap between matrix theory and linear
algebra.
(cid:2) Chapters 7 and 8 are more advanced, where we introduce all the
necessary ideas concerningeigenvaluesand eigenvectorsand thetheory of
symmetricandorthogonalmatrices.
Oneoftheaspectsthatshouldmakethistextbookusefulforstudentsis
the presence of exercises at the end of each chapter. We did choose these
exercises very carefully to illustrate the main ideas. Since some of them
are taken (with some modifications) from recently published papers, it is
possiblethattheseexercises appearfor thefirsttimein atextbook.Allthe
exercisesareprovidedwithdetailedsolutionsandineachsolution,werefer
tothemaintheoremsinthetextwhennecessary,sostudentscanseethemain
tools used in the solution.In addition all the mainideasinthisbookcome
viii Preface
with illustrating examples. We did strive to choose solutions and proofs
that are elegant and short. We also tried to make this textbook in about
400 pages by focusing on the main ideas, so that students will be able to
easily and quickly understand things. In addition, we tried to maintain a
balance between the theory of matrices and the one of vector spaces and
lineartransformations.
Thisbookcanbeusedasatextbookforafirstcourseinlinearalgebrafor
undergraduatestudentsinalldisciplines.Itcanbealsousedasabookletfor
graduatestudents,allowingto acquire someconcepts, examples,and basic
results.Itisalsosuitableforthosestudentswhoarelookingforsimple,easy,
andcleartextbookthatsummarizesthemainideasoflinearalgebra.Finally
itisalsointendedforthosestudentswhoareinterestedinrigorousproofsof
themaintheoremsinlinearalgebra.Webelievethatifagoodstudentuses
thisbook,thenshe(he)canreadandlearnthebasicsoflinearalgebraonher
(his)own.
WewouldliketothankSalimA.Messaoudi(fromKFUPM)forvaluable
suggestionsand corrections which improved the contents of some parts of
thisbook.WealsothankSofianeBouarroudj(fromNYUAbuDhabi)forthe
many discussions that we have had about the proofs of some theorems of
linearalgebra.
AbuDhabi,UnitedArabEmirates BelkacemSaid-Houari
October06,2016
ix
Contents
1 MatricesandMatrixOperations................................ 1
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1.1 SystemsofLinearEquations........................................ 1
1.1.1 TheGroupofMatrices............................................... 11
1.1.2 MultiplicationofMatrices ........................................... 12
1.2 SquareMatrices.................................................... 19
1.2.1 TheRingofSquareMatrices......................................... 19
1.2.2 TheIdentityMatrix.................................................. 21
1.2.3 InverseMatrices..................................................... 22
1.2.4 DiagonalMatrices................................................... 30
1.2.5 TriangularMatrices.................................................. 35
1.2.6 Trace................................................................ 37
1.3 SolvingLinearSystemswithElementaryRowOperations .......... 39
1.3.1 TheGauss–JordanEliminationMethod .............................. 41
1.4 TheMatrixTransposeandSymmetricMatrices ..................... 49
1.4.1 TransposeofaMatrix................................................ 50
1.4.2 SymmetricMatrices................................................. 52
1.5 Exercises ........................................................... 53
2 Determinants..................................................... 69
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2.1 Introduction........................................................ 69
2.2 DeterminantsbyCofactorExpansion............................... 70
2.3 PropertiesoftheDeterminants .................................... 76
2.4 EvaluatingDeterminantsbyRowReduction........................ 79
2.4.1 DeterminantTestforInvertibility .................................... 89
2.5 TheAdjointofaSquareMatrix ..................................... 91
2.5.1 Cramer’sRule ....................................................... 99
2.6 Exercises ........................................................... 104
3 EuclideanVectorSpaces......................................... 121
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3.1 Introduction........................................................ 121
3.2 VectorAdditionandMultiplicationbyaScalar ..................... 121
3.2.1 VectorAddition..................................................... 121
3.2.2 MultiplicationofaVectorbyaScalar................................. 123
3.2.3 VectorsinCoordinateSystems....................................... 123
3.2.4 LinearCombination................................................. 127
3.3 Norm,DotProduct,andDistanceinRn ............................. 128
3.3.1 TheNormofaVectorinRn .......................................... 128
3.3.2 DistanceinRn....................................................... 131
3.3.3 TheDotProductofTwoVectorsinRn................................ 132
x Contents
3.4 OrthogonalityinRn................................................. 141
3.4.1 OrthogonalProjectionsinRn ........................................ 142
3.5 Exercises............................................................ 145
4 GeneralVectorSpaces ........................................... 159
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4.1 DefinitionandExamples............................................ 159
4.2 PropertiesofVectorSpaces......................................... 161
4.3 Subspaces .......................................................... 163
4.3.1 DirectSumDecomposition.......................................... 170
4.4 LinearIndependence ............................................... 172
4.5 BasesofVectorSpaces.............................................. 175
4.6 DimensionofaVectorSpace........................................ 178
4.6.1 DimensionofaSubspace............................................ 182
4.6.2 ConstructionofaBasis............................................... 182
4.7 Exercises............................................................ 183
5 LinearTransformations.......................................... 199
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5.1 DefinitionandExamples ........................................... 199
5.2 FundamentalPropertiesofLinearTransformations................. 201
5.2.1 TheKernelandtheImageofaLinearTransformation ................ 204
5.3 IsomorphismofVectorSpaces...................................... 211
5.4 Exercises............................................................ 215
6 LinearTransformationsandMatrices .......................... 227
BelkacemSaid-Houari
6.1 DefinitionandExamples ........................................... 227
6.2 ChangeofBasisandSimilarity...................................... 234
6.3 RankofMatrices.................................................... 241
6.3.1 SomePropertiesoftheRankofaMatrix.............................. 243
6.4 MethodsforFindingtheRankofaMatrix........................... 246
6.4.1 TheMethodofElementaryRowandColumnOperations ............. 246
6.4.2 TheMethodofMinorsforFindingtheRankofaMatrix ............... 251
6.5 Exercises............................................................ 255
7 EigenvaluesandEigenvectors................................... 269
BelkacemSaid-Houari
7.1 Definitions.......................................................... 269
7.2 PropertiesofEigenvaluesandEigenvectors ........................ 272
7.3 EigenvaluesandEigenvectorsofaMatrix........................... 277
7.4 Diagonalization..................................................... 290
7.4.1 SpectrumofSymmetricMatrices .................................... 296