Linear Algebra and Geometry PDF
Preview Linear Algebra and Geometry
Linear Algebra and Geometry Igor R. Shafarevich (cid:2) Alexey O. Remizov Linear Algebra and Geometry Translated by David Kramer and Lena Nekludova IgorR.Shafarevich AlexeyO.Remizov SteklovMathematicalInstitute CMAP RussianAcademyofSciences ÉcolePolytechniqueCNRS Moscow,Russia PalaiseauCedex,France Translators: DavidKramer Lancaster,PA,USA LenaNekludova Brookline,MA,USA TheoriginalRussianeditionwaspublishedas“Linejnayaalgebraigeometriya”byFizmatlit, Moscow,2009 ISBN978-3-642-30993-9 ISBN978-3-642-30994-6(eBook) DOI10.1007/978-3-642-30994-6 SpringerHeidelbergNewYorkDordrechtLondon LibraryofCongressControlNumber:2012946469 MathematicsSubjectClassification(2010): 15-01,51-01 ©Springer-VerlagBerlinHeidelberg2013 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) Preface Thisbookistheresultofaseriesoflecturesonlinearalgebraandthegeometryof multidimensionalspacesgiveninthe1950sthrough1970sbyIgorR.Shafarevich attheFacultyofMechanicsandMathematicsofMoscowStateUniversity. Notesforsomeoftheselectureswerepreservedinthefacultylibrary,andthese were used in preparing this book. We have also included some topics that were discussedinstudentseminarsatthetime.Allthematerialincludedinthisbookis theresultofjointworkofbothauthors. We employ in this book some results on the algebra of polynomials that are usually taught in a standard course in algebra (most of which are to be found in Chaps. 2 through 5 of this book). We have used only a few such results, without proof: the possibility of dividing one polynomial by another with remainder; the theoremthatapolynomialwithcomplexcoefficientshasacomplexroot;thatevery polynomialwithrealcoefficientscanbefactoredintoaproductofirreduciblefirst- andsecond-degreefactors;andthetheoremthatthenumberofrootsofapolynomial thatisnotidenticallyzeroisatmostthedegreeofthepolynomial. To provide a visual basis for this course, it was preceded by an introductory courseinanalyticgeometry,towhichweshalloccasionallyrefer.Inaddition,some topicsandexamplesareincludedinthisbookthatarenotreallypartofacoursein linearalgebraandgeometrybutareprovidedforillustrationofvarioustopics.Such itemsaremarkedwithanasteriskandmaybeomittedifdesired. For the convenience of the reader, we present here the system of notation used in this book. For vector spaces we use sans serif letters: L,M,N,...; for vectors, weuseboldfaceitalics:x,y,z,...;forlineartransformations,weusecalligraphic letters: A,B,C,...; and for the corresponding matrices, we use uppercase italic letters:A,B,C,.... Acknowledgements TheauthorsaregratefultoM.I.Zelinkin,D.O.Orlov,andYa.V.Tatarinovforread- ingpartsofanearlierversionofthisbookandmakinganumberofusefulsugges- v vi Preface tionsandremarks.Theauthorsarealsodeeplygratefultooureditor,S.Kuleshov, who gave the manuscript a very careful reading. His advice resulted in a number of important changes and additions. In particular, some parts of this book would not have appeared in their present form had it not been for his participation in thisproject.Wewouldalsoliketoofferourheartythankstothetranslators,David KramerandLenaNekludova,fortheirEnglishtranslationandinparticularforcor- rectinga numberof inaccuraciesandtypographicalerrors that werepresent in the Russianeditionofthisbook. Contents 1 LinearEquations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 LinearEquationsandFunctions . . . . . . . . . . . . . . . . . . . 1 1.2 GaussianElimination . . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Examples* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 MatricesandDeterminants . . . . . . . . . . . . . . . . . . . . . . . 25 2.1 DeterminantsofOrders2and3 . . . . . . . . . . . . . . . . . . . 25 2.2 DeterminantsofArbitraryOrder . . . . . . . . . . . . . . . . . . 30 2.3 PropertiesthatCharacterizeDeterminants. . . . . . . . . . . . . . 37 2.4 ExpansionofaDeterminantAlongItsColumns . . . . . . . . . . 39 2.5 Cramer’sRule . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.6 Permutations,SymmetricandAntisymmetricFunctions . . . . . . 44 2.7 ExplicitFormulafortheDeterminant . . . . . . . . . . . . . . . . 50 2.8 TheRankofaMatrix . . . . . . . . . . . . . . . . . . . . . . . . 53 2.9 OperationsonMatrices . . . . . . . . . . . . . . . . . . . . . . . 60 2.10 InverseMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3 VectorSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.1 TheDefinitionofaVectorSpace . . . . . . . . . . . . . . . . . . 79 3.2 DimensionandBasis . . . . . . . . . . . . . . . . . . . . . . . . 86 3.3 LinearTransformationsofVectorSpaces . . . . . . . . . . . . . . 101 3.4 ChangeofCoordinates. . . . . . . . . . . . . . . . . . . . . . . . 107 3.5 IsomorphismsofVectorSpaces . . . . . . . . . . . . . . . . . . . 112 3.6 TheRankofaLinearTransformation . . . . . . . . . . . . . . . . 118 3.7 DualSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.8 FormsandPolynomialsinVectors . . . . . . . . . . . . . . . . . 127 4 LinearTransformationsofaVectorSpacetoItself . . . . . . . . . . 133 4.1 EigenvectorsandInvariantSubspaces . . . . . . . . . . . . . . . . 133 4.2 ComplexandRealVectorSpaces . . . . . . . . . . . . . . . . . . 142 4.3 Complexification . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 4.4 OrientationofaRealVectorSpace . . . . . . . . . . . . . . . . . 154 vii viii Contents 5 JordanNormalForm . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.1 PrincipalVectorsandCyclicSubspaces . . . . . . . . . . . . . . . 161 5.2 JordanNormalForm(Decomposition) . . . . . . . . . . . . . . . 165 5.3 JordanNormalForm(Uniqueness) . . . . . . . . . . . . . . . . . 169 5.4 RealVectorSpaces. . . . . . . . . . . . . . . . . . . . . . . . . . 173 5.5 Applications* . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 6 QuadraticandBilinearForms . . . . . . . . . . . . . . . . . . . . . . 191 6.1 BasicDefinitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 6.2 ReductiontoCanonicalForm . . . . . . . . . . . . . . . . . . . . 198 6.3 Complex,Real,andHermitianForms . . . . . . . . . . . . . . . . 204 7 EuclideanSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 7.1 TheDefinitionofaEuclideanSpace . . . . . . . . . . . . . . . . 213 7.2 OrthogonalTransformations . . . . . . . . . . . . . . . . . . . . . 223 7.3 OrientationofaEuclideanSpace*. . . . . . . . . . . . . . . . . . 230 7.4 Examples* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233 7.5 SymmetricTransformations . . . . . . . . . . . . . . . . . . . . . 245 7.6 ApplicationstoMechanicsandGeometry* . . . . . . . . . . . . . 255 7.7 Pseudo-EuclideanSpaces . . . . . . . . . . . . . . . . . . . . . . 265 7.8 LorentzTransformations . . . . . . . . . . . . . . . . . . . . . . . 275 8 AffineSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 8.1 TheDefinitionofanAffineSpace . . . . . . . . . . . . . . . . . . 289 8.2 AffineSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 8.3 AffineTransformations . . . . . . . . . . . . . . . . . . . . . . . 301 8.4 AffineEuclideanSpacesandMotions . . . . . . . . . . . . . . . . 309 9 ProjectiveSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 9.1 DefinitionofaProjectiveSpace . . . . . . . . . . . . . . . . . . . 319 9.2 ProjectiveTransformations . . . . . . . . . . . . . . . . . . . . . 328 9.3 TheCrossRatio . . . . . . . . . . . . . . . . . . . . . . . . . . . 335 9.4 TopologicalPropertiesofProjectiveSpaces* . . . . . . . . . . . . 339 10 TheExteriorProductandExteriorAlgebras . . . . . . . . . . . . . 349 10.1 PlückerCoordinatesofaSubspace . . . . . . . . . . . . . . . . . 349 10.2 ThePlückerRelationsandtheGrassmannian . . . . . . . . . . . . 353 10.3 TheExteriorProduct . . . . . . . . . . . . . . . . . . . . . . . . . 358 10.4 ExteriorAlgebras* . . . . . . . . . . . . . . . . . . . . . . . . . . 367 10.5 Appendix* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374 11 Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 11.1 QuadricsinProjectiveSpace . . . . . . . . . . . . . . . . . . . . 385 11.2 QuadricsinComplexProjectiveSpace . . . . . . . . . . . . . . . 394 11.3 IsotropicSubspaces . . . . . . . . . . . . . . . . . . . . . . . . . 398 11.4 QuadricsinaRealProjectiveSpace . . . . . . . . . . . . . . . . . 410 11.5 QuadricsinaRealAffineSpace . . . . . . . . . . . . . . . . . . . 414 11.6 QuadricsinanAffineEuclideanSpace . . . . . . . . . . . . . . . 425 11.7 QuadricsintheRealPlane* . . . . . . . . . . . . . . . . . . . . . 428 Contents ix 12 HyperbolicGeometry . . . . . . . . . . . . . . . . . . . . . . . . . . 433 12.1 HyperbolicSpace* . . . . . . . . . . . . . . . . . . . . . . . . . . 434 12.2 TheAxiomsofPlaneGeometry* . . . . . . . . . . . . . . . . . . 443 12.3 SomeFormulasofHyperbolicGeometry* . . . . . . . . . . . . . 454 13 Groups,Rings,andModules . . . . . . . . . . . . . . . . . . . . . . . 467 13.1 GroupsandHomomorphisms . . . . . . . . . . . . . . . . . . . . 467 13.2 DecompositionofFiniteAbelianGroups . . . . . . . . . . . . . . 475 13.3 TheUniquenessoftheDecomposition . . . . . . . . . . . . . . . 481 13.4 FinitelyGeneratedTorsionModulesoveraEuclideanRing* . . . . 484 14 ElementsofRepresentationTheory . . . . . . . . . . . . . . . . . . . 497 14.1 BasicConceptsofRepresentationTheory . . . . . . . . . . . . . . 497 14.2 RepresentationsofFiniteGroups . . . . . . . . . . . . . . . . . . 503 14.3 IrreducibleRepresentations . . . . . . . . . . . . . . . . . . . . . 508 14.4 RepresentationsofAbelianGroups . . . . . . . . . . . . . . . . . 511 HistoricalNote . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 515 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 Preliminaries Inthisbookweshalluseanumberofconceptsfromsettheory.Theseideasappear inmostmathematicscourses,andsotheywillbefamiliartosomereaders.However, weshallrecallthemhereforconvenience. SetsandMappings Asetisacollectionofarbitrarilychosenobjectsdefinedbycertainpreciselyspeci- fiedproperties(forexample,thesetofallrealnumbers,thesetofallpositivenum- bers, the set of solutions of a given equation, the set of points that form a given geometric figure, the set of wolves or trees in a given forest). If a set consists of a finite number of elements, then it is said to be finite, and if not, it is said to be infinite.Weshallemploystandardnotationforcertainimportantsets,denotingthe setofnaturalnumbersbyN,thesetofintegersbyZ,thesetofrationalnumbersby Q,thesetofrealnumbersbyR,andthesetofcomplexnumbersbyC.Thesetof naturalnumbersnotexceedingagivennaturalnumbern,thatis,thesetconsisting of 1,2,...,n,willbedenotedby N .Theobjectsthatmakeupasetarecalledits n elementsorsometimespoints.If x isanelementoftheset M,thenweshallwrite x ∈M. If we need to specify that x in not an element of M, then we shall write x∈/M. AsetS consistingofcertainelementsofthesetM (thatis,everyelementofthe set S is also an element of the set M) is called a subset of M. We write S ⊂M. Forexample,N ⊂Nforarbitraryn,andlikewise,wehaveN⊂Z,Z⊂Q,Q⊂R, n andR⊂C.AsubsetofM consistingofelementsx ∈M (wheretheindexα runs α overagivenfiniteorinfiniteset)willbedenotedby{x }.Itisconvenienttoinclude α among the subsets of a set M the set that contains no elements at all. We call this settheemptysetanddenoteitby∅. LetM andN betwoarbitrarysets.Thecollectionofallelementsthatbelongsi- multaneouslytobothM andN iscalledtheintersectionofM andN andisdenoted byM∩N.IfwehaveM∩N =∅,thenwesaythatthesetsM andN aredisjoint. xi
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