lecture+1+Review+of+basic+concepts+and+facts+in+linear.ppt

Reviewofbasicconceptsandfactsinlinearalgebra,MatrixAnalysisHITSZInstructor:ZijunLuoFall2012,guideline,Prerequisite:Text:MatrixAnalysis,RogerA.Horn,CharlesR.Johnson,CambridgeUniversityPress,Reprintedition(February23,1990).ISBN0521386322,575pHomeworkassignment:weeklyExam:finalexam(openbook)onJan.14,2013(temporary)Content:vectorspace,norms,eigenvalues,unitarymatrix,Hermitianmatrix,matrixfactorization,canonicalform,nonnegativematrix,Definition:VectorSpace(V,+;F)Avectorspace(overafieldF)consistsofasetValongwith2operations+ands.t.Forthevectoraddition+:v,w,uVv+wV(Closure)v+w=w+v(Commutativity)(v+w)+u=v+(w+u)(Associativity)0Vs.t.v+0=v(Zeroelement)vVs.t.vv=0(Inverse)(2)Forthescalarmultiplication:v,wVanda,bF,avV(Closure)(a+b)v=av+bv(Distributivity)a(v+w)=av+aw(ab)v=a(bv)=abv(Associativity)1v=v(Scalaridentityofmultiplication),Expressionofforce,velocity,gradient,SubspaceAsetUisasubspaceofavectorspaceVifEveryelementofUisinV,andUisavectorspace.,LinearCombinations,Considerasetofvectorsv1,.,vnandasetofscalarsa1,anAlinearcombinationofthevectorsisa1v1+a2v2+anvn,Remark:Vectorspace=Collectionoflinearcombinationsofvectors.,Definition:SpanLetS=s1,sn|sk(V,+,R)beasetofnvectorsinvectorspaceV.ThespanofSisthesetofalllinearcombinationsofthevectorsinS,i.e.,with,Lemma:Thespanofanysubsetofavectorspaceisasubspace.,Proof:,LetS=s1,sn|sk(V,+,R),and,QED,Note:spanSisthesmallestvectorspacecontainingallmembersofS.,Example:,Proof:Theproblemistantamounttoshowingthatforallx,yR,uniquea,bRs.t.,i.e.,hasauniquesolutionforarbitraryx*Subsetofal.i.setisl.i.;*L.i.vectorscanbeaddedtoformabasis;*Everybasishasthesamenumberofvectors;*Eachvectorhasauniquebasis-representation;*EveryinnerproducthastheCauchy-Schwarzinequality.*Everyinnerproductcanbeusedtodefineavectorno