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1、LinearAlgebraDoneWrongSergeiTreilDepartmentofMathematics,BrownUniversityCopyrightcSergeiTreil,2004,2009PrefaceThetitleofthebooksoundsabitmysterious.Whyshouldanyonereadthisbookifitpresentsthesubjectinawrongway?Whatisparticularlydonewrong"inthebook?Beforeansweringthesequestions,letme rstdescri
2、bethetargetaudi-enceofthistext.ThisbookappearedaslecturenotesforthecourseHonorsLinearAlgebra".Itsupposedtobea rstlinearalgebracourseformath-ematicallyadvancedstudents.Itisintendedforastudentwho,whilenotyetveryfamiliarwithabstractreasoning,iswillingtostudymorerigorousmathematicsthatispresente
3、dinacookbookstyle"calculustypecourse.Besidesbeinga rstcourseinlinearalgebraitisalsosupposedtobea rstcourseintroducingastudenttorigorousproof,formalde nitions
4、inshort,tothestyleofmoderntheoretical(abstract)mathematics.Thetargetaudienceexplainstheveryspeci cblendofelementaryideasandconcreteexa
5、mples,whichareusuallypresentedinintroductorylinearalgebratextswithmoreabstractde nitionsandconstructionstypicalforadvancedbooks.Anotherspeci cofthebookisthatitisnotwrittenbyorforanalge-braist.So,Itriedtoemphasizethetopicsthatareimportantforanalysis,geometry,probability,etc.,anddidnotincludeso
6、metraditionaltopics.Forexample,Iamonlyconsideringvectorspacesoverthe eldsofrealorcom-plexnumbers.Linearspacesoverother eldsarenotconsideredatall,sinceIfeeltimerequiredtointroduceandexplainabstract eldswouldbebetterspentonsomemoreclassicaltopics,whichwillberequiredinotherdis-ciplines.Andlater,
7、whenthestudentsstudygeneral eldsinanabstractalgebracoursetheywillunderstandthatmanyoftheconstructionsstudiedinthisbookwillalsoworkforgeneral elds.iiiivPrefaceAlso,Itreatonly nite-dimensionalspacesinthisbookandabasisalwaysmeansa nitebasis.Thereasonisthatitisimpossibletosaysome-thingnon-trivial
8、aboutin nite-dimensionalspaceswithoutintroducingcon-vergence,norms,completenessetc.,i.e.thebasicsoffunctionalanalysis.Andthisisde nitelyasubjectforaseparatecourse(text).So,Idonotconsiderin niteHamelbaseshere:theyarenotneededinmostapplica-tion
