Table Of Content

LINEARALGEBRAANDGEOMETRY TranslatedfromtheDutchedition byA.VANDERSuns LINEAR ALGEBRA AND GEOMETRY NICOLAASH. KUIPER Pro/assayofMathematics Um'unsiWofAmsterdam SECONDREVISEDEDITION "NM. NORTH-HOLLANDPUBLISHINGCOMPANY—AMSTERDAM Nopartofthisbookmaybereproducedinanyform byprint,photoprint,microfilmoranyothermeans withoutwrittenpermissionfromthepublisher 15!Edition1962 andprinting1963 andrevisededition1965 PUBLISHERS: NORTH-HOLLANDPUBLISHINGC0,, AMSTERDAM sou;DISTRIBUTOR?FORU.S.A.: INTERSCIENCEPUBLISHERS INC., NEWYORK PRINTEDINTHENETHERLANDS PREFACE Thesubjectwhichwasusuallydesignatedbythenamesanalytic geometry,affinegeometryandprojectivegeometry,hasgradually grownintoanewsubjectunderthegreatunifyingandstimulating influenceofalgebra.Herelinearalgebraplaysadominantroledue toitsuse asatechnique andalsobecausemanyoftheessential ideasofthesubjectbelongtolinearalgebra. Inthisbookwefollowthistrend.Wetrytopenetrateoursubject- matterwiththemodernalgebraictoolsthatareusefulinallmathe— matics.Howeverithasbeenouraimatthesametimetopreserve theaestheticallysatisfyinggeometricalflavourofthesubject.We haveincludedmanygeometricaltheorems,someelementary,some importantinafield—freeaxiomaticapproach (likethetheoremof Desargues). Anotherspecialfeatureinourtreatmentisthatwehaveempha- sizedtheideaoffunctionsonspaces,inallthosecaseswhereitis customarytodealwiththeequations,curvesandsurfaces,which representthelevel-setsofthesefunctions.Ourmethodisveryfertile, ascanbeseenfromthetreatmentofquadraticsurfacesandthe invarianceofcross-ratiounderprojection. Ingeneralwehavetriedtokeepourtreatmentcoordinate-free. In particularwehave not followed the custom ofreplacingany spacebyasetofcoordinates,andthenforgettingaboutthespaceas soon aspossible. Inthisspiritwehandlehomomorphism (linear mappings)beforeconsideringmatrices.Whileusingcoordinateswe havebeenconsciousofthefactthatcoordinatesalsoarefunctions. Asfaraslinearalgebraisconcernedthemainresulttowhichthis bookleadsisthe classificationofthe endomorphismofcomplex (C)andreal(R)finitedimensionalvectorspaces, (Jordan normal VI PREFACE form).Wealsogivetheclassificationofquadraticandhermitian functions(orforms)oneuclideanandunitaryvectorspaces. Asfarasgeometryisconcernedwedealwitheuclideanandaffine spaces, particularythose ofdimensiontwo andthree, withtheir motionsandaffinities,andlinearandquadraticfunctions. Alongchapter(Ch.20)isdevotedtoprojectivegeometry,while some attentionispaidto finite planes (Ch.8), non—euclideange- ometry(Ch.21)andthetopologyofsomeveryelementaryvarieties (Ch.22).InCh. 16thetheoryofleastsquaresandcorrelationcoef- ficientsismentionedastheapproachfromourpointofViewandin ourformulationseemsclarifyingandisnotgenerallyknown. WhilepreparingtheDutcheditionofthisbookandalsoduring the preparation of this translation we have had the benefits of criticalremarks, suggestions andhelp fromProfessorF. van der Blij,Mr.M.Keuls,ProfessorT.A.Springer,Mr.L. R. Verdooren, andothers.Weareverygratefulforthishelp. ThetranslatorDr.A.vanderSluishasnotrestrictedhisworkto afaithfultranslation,but,incooperationwiththeauthor,hehas improvedonthe Dutch editioninviewofhisown criticism and thoseofothers.TheauthorexpresseshisextremegratitudetoDr. A.vanderSluis.Thetranslator,inhisturn,wishesto express his warm appreciation to the author for theopen-mindedness and cooperationwithwhichhewelcomedanysuggestion. Dr.T.J.WillrnorefromLiverpoolUniversityreadthetextand made several improvements, particularly those of a linguistic nature.Wethankhimwarmlyforthisworkandhisinterest. Finallyweliketothankthepublisherforhiscontributiontothe bookandforthepleasantcooperation. Wageningen,September1961. N.H.KUIPER A.VANDERSuns CONTENTS CHAPTER VECTORSINTEEPLANEANDINSPACE . . . . . . . . . . 2. SUBSET.PRODUCTSET,RELATIONANDMAPPING ....... 91.59:. TSTHOHEMEEnP-AFDRUIMANMEDENATSMRIEOICNNTARALELPVRTEEHCSETEOONRRTEAMSTPSIAOCN.EO.VF.”A..LI.N..E.....1.... ... ... ... -.. ThedualvectorspaceV" ..... . .......... m. FIRSTDEGREEFUNCTIONSON,ANDLINEARVARIETIESINA. . u. LINEAR'FUNCTIONSANDLINESINA2ANDA".APPLICATIONS. Cross-ratio................... . . . . Harmonicseparation.................. can.. HAOFMINOIMTOERAPFHFISINMESOPLFAVNEECT.OR.S.PA.C.ES...... .................. ThevectorspaceHom(A,B) . . ........... . Composition(multiplication)ofhomomorphism...... Thedualhomomorphismofthedualvectorspaces..... . MATRICES ....................... SETS0FLINEAREQUATIONS . . . . ........... . FUNCTIONSOFSEVERALVARIABLES. DETERMINANT ..... . APPLICATIONSOFDETERMINANTS.VOLUME...... . . . QUADRATICANDSYMMETRICBILINEARFUNCTIONS. . . . A.Functionsonavectorspace. . . . . . . . . . . . . An.Hennitianfunctions................ B. Functionsonarealaffinespace. . . . . . . . . . . . .EUCLIDEANSPACE....-........’..... Unitaryvectorspaoe. . SOMEAPPLICATIONSIN STATISTICS . . . . . . . . . . . I. Methodofleastsquares linearadjustment,regression. . . II. Thecorrelationcoefficient . . . . . . . . . . . . . . . CLASSIFICATION OF ENDOMORPHISMS . . . . . . . . . . . Classificationofendomorphisms(complexnumbers) . . . . Endomorphismsofrealvectorspaces........ . . . Symmetricendomorphismsandquadraticfunctionsonaeu- dideanvectorspace.................. VIII CONTENTS Orthogonalendomorphism ............... 173 Hermitianendomorphismsandhermitianfunctionsonauni- taryspace......... . ........... 176 18. QvADRA'IICFUNCTIONSONANDQUADRA‘HCVARIETXESINEU- CLIDEANSPACES .................... 179 Investigationofagivenquadraticvariety......... 183 19. MOTIONSANDAFFINITIES . ............... 190 Motions........................ 199 Classificationofmotions................. 195 Motionintheeuclideanplaneasbasicnotion ....... 199 Affinitiesinrealspaces ................. 201 Someconstructionswithplaneaffinities.......... 204 20. PROJECTIVEGEOMETRY ................. 209 PointsatinfinityofanaffineplaneA2 .......... 209 Projectiveclassificationofquadrics(overCandR) ..... 221 Classificationofcollineations(overCandR)........ 224 21. NON-EUCLIDEANPLANES................. 237 Thehyperbolicplane.................. 237 Theellipticplane.................... 244 22. SOMETOPOLOGICALREMARKS............... 248 HINTSANDANSWERSTOTHEPROBLEMSINCHAPTER3—21 ..... 255 INDEX........................... 279 LISTOFSYMBOLS...................... 285 l. VECTORS IN THE PLANE AND IN SPACE Intheordinaryplane(orinordinaryspace)wetakeafixedpoint 0, thatwe callthe-origin. We consider arrowsinthe plane. An arrowcan becharacterizedbyitsinitialpoint and itsendpoint. Thewordarrowisthereforeusedassynonymfor”orderedpairof points”,thefirstpointofapairbeingtheinitialpoint,thesecond theendpointofthearrow.Anarrowwith0asinitialpointwillbe calledavector1). Aspecialvectoristhatwith0asinitialaswellasendpoint;itis specialsinceitcannotbedrawnasanordinaryarrow.Itiscalled thezerovectoranddesignatedby0. Thereisaone»to-onerelationshipbetweenpointsoftheplaneand vectors; indeed, toeachpoint there corresponds onevectorwith thatpointasendpoint,andconversely,toanyvectortherebelongs anendpoint. To any arrow BC (cf. fig. 1.1) there corresponds exactly one vector (thearrowwith0 asinitialpoint) whichcanbeobtained fromBCbydisplacingitparalleltoitselfsothat 0becomesthe initialpoint.Thevector0Asoobtainedmaythusberepresentedby thearrow30 Thevector0A has manysuchrepresentatives, of course. Sometimeswe shallsay "thevectorBC" andmean“the VectorrepusmtedbyBC". Theset ofall arrowsintheplane (in space) withinitialpoint 0 is called a 2—dimnsional vector space (a. 3<dimensional vector space). Furthermore the following algebraic conventions will be observed: 1) Another(equivalent)definitionwhich,however,isslightlylessrealistic tothebeginner,is:aVectorisacompleteset(Le.asetthatcannotbeex- tended)ofarrows,eachofwhichcanbeobtainedfromanyotheroneby translation(aparalleldisplacement). 2 VECTORS IN THE PLANE AND IN SPACE A1.Addition. To any pair of vectors a=0A, b=OB there correspondsavector0=0Cinthefollowingway:thearrowBC mustbearepresentativeofa,orequivalently,thearrowACmust §—_£ I I / ,’ / / ‘0 ~#0 °lI «,1/ 2a O A D Fig.1.x bearepresentativeofb,or:Cisthefourthvertexofaparallelogram withsides0Aand03(of.fig. 1.2.Thelastformulation,knownas theparallelogram—constmction,isnotunambiguousif0,AandB are collinear). Thisvectorciscalledthesumofaandb, andis denotedby a+b. Fig.1.2 A2.MultiflimtionbyarealnumberA.Toanypairconsistingofa vector a=0A and a real number 2 there corresponds a vector (1=0D,calledtheproductofAanda,denotedbyla.inthefollow- ingway:0,A andDlieona.lineand lengthofOD=|1|~lengthof0A, whilstA andDwillbeonthesameoronoppositesidesof0de- pendingonwhetherI.ispositiveornegative.Illmeanstheabsolute valueof1,i.e.thenon-negativeoneofthenumbersI.and—l;for example [5|=5, l—7]=7, |Ol=0‘(cf. fig. 1.] with7.=2). The