UChicago Notes - Linear Algebra

UChicago Notes - Linear Algebra

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1、LinearAlgebraLectureNotesinprogressL´aszl´oBabaiVersion:November11,2007Thesenotesarebasedonthe“apprenticecourse”andthe“discretemathematics”coursegivenbytheauthorattheSummer2007REUoftheDepartmentofMathematics,UniversityofChicago.TheauthorisgratefultothescribesSundeepBalajiandShawnDrenning(others

2、cribeswillbeacknowledgedastheirpartsgetincorporatedinthetext).1Basicstructures1.1GroupsAgroupisasetGendowedwithabinaryoperation,usuallycalledadditionormultiplication,satisfyingthefollowingaxioms(writteninmultiplicativenotation):(a)(∀a,b∈G)(∃!ab∈G)(operationuniquelydefined)(b)(∀a,b,c∈G)((ab)c=a(b

3、c))(associativity)(c)(∃e∈G)(∀a∈G)(ea=ae=a)(identityelement)(d)(∀a∈G)(∃b∈G)(ab=ba=e)(inverses)Inadditivenotation,wepostulate(a)(∀a,b∈G)(∃!a+b∈G)(operationuniquelydefined)(b)(∀a,b,c∈G)((a+b)+c=a+(b+c))(associativity)(c)(∃e∈G)(∀a∈G)(e+a=a+e=a)(identityelement)(d)(∀a∈G)(∃b∈G)(a+b=b+a=e)(inverses)1Th

4、emultiplicativeidentityisusuallydenotedby“1,”theadditiveidentityby“0.”Themultiplicativeinverseofaisdenotedbya−1;theadditiveinverseby(−a).Thegroupiscommutativeorabelianifitsatisfies(∀a,b∈G)(ab=ba)(or(∀a,b∈G)(a+b=b+a)intheadditivenotation).Theadditivenotationiscustomarilyreservedforabeliangroups.E

5、xample1.1.1.(Z,+)(theadditivegroupofintegers),(Zn,+)(theadditivegroupofmod-ulonresidueclasses),thegenerallineargroupGL2(p)(2×2matricesoverZpwithnonzerodeterminant(nonzeromodpwherepisprime))undermatrixmultiplication,thespeciallineargroupSL2(p)(thesubgroupofGL2(p)consistingofthosematriceswithdete

6、rminant=1(modp))Exercise1.1.2.Ifpisaprime(Z×,·)isagroup.HereZ×isthesetofnon-zeroresidueppclassesmodulop.Theorderofagroupisthenumberofelementsofthegroup.Forinstance,theorderof(Z,+)isn;theorderof(Z×,·)isp−1;theorderof(Z,+)isinfinite.Notethattheordernpofagroupisatleast1sinceithasanidentityelement.T

7、heidenityelementaloneisagroup.Exercise1.1.3.CalculatetheorderofthespeciallineargroupSL2(p).(Giveaverysimpleexactformula.)1.2FieldsInformally,afieldisasetFtogetherwithtwobinaryoperations,additionandmultiplication,sothatalltheusualid

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