第1題Iftherowreducedechelonformofinvolvesfreevariables,thenthesystemwillhaveinfinitelymanysolutions.()第2題Everyhomogeneouslinearsystemisconsistent.()第3題Anmatrixisnonsingularifandonlyifthereducedrowechelonformofis(theidentitymatrix).()第4題Ifisnonsingular,thencanbefactoredintoaproductofelementarymatrices.()第5題Ifandarenonsingularmatrices,thenisalsononsingularand.()第6題If,thenmustbeequaltoeitheror()第7題Ifandarematrices,then.()第8題Ifand

CisnotO(thezeromatrix),then.()第9題If,then.()第10題Ifisamatrixand,thenmustbesingular.()第11題LetbeasystemoflinearequationsinunknownsandsupposethatandarebothsolutionsandThesystemhas-solutions.Thematrixis_______.AinfinitelymanynonsingularBinfinitelymanysingularCtwononsingularDtwosingular第12題Letandbe10×10matricesthatarepartitionedintosubmatricesasfollowsIfisa6×5matrix,andisak×rmatrix,inordertomaketheblockmultiplicationoftimespossible,andtherangeofare______.A6B6C5D5第13題Letbeallsquarematrices.Theremustbe________.A(AB)T=ATBTBCD第14題Letbeamatrix,then___________.AIfhasonlytrivialsolution,thenmusthaveauniquesolution.BIfhasanon-trivialsolution,thenmusthaveinfinitelymanysolutions.CIfhasauniquesolution,thenmusthaveauniquesolution.DIfhasanon-trivialsolution,thenmusthaveanon-trivialsolution.第15題Letbeallsquarematrices,if,then_________.AorBCD第1題()第2題()第3題()第4題()第5題implies.()第6題()第7題Atriangularmatrixisnonsingularifandonlyifitsdiagonalentriesareallnonzero.()第8題IfxandyaretwodistinctsvectorsinRn

andAx=Ay,thendet(A)=0.第9題Ifandarerowequivalentmatrices,thentheirdeterminantsareequal.()第10題If,but(wheredenotesthezeromatrix)forsomepositiveinteger,thenmustbesingular.()第11題Ifisanonsingularmatrix,thenthatis________and_______.

Anonsingular

>0Bnonsingular<0Csingular

>0Dsingular

<0第12題Letbeamatrix,thedeterminantofis,and

A?theadjointmatrixofA,then

|(3A)?1+2A?|=__________.A1B3C6D9第13題Letthen=__________.A

BCD第14題arematrices,let,,and,,thedeterminantof

(A+B)=_________.A1B3C6D9第15題Weknowthathasonlyauniquesolution,thensatisfies________.ABCorDand

Homework3第1題IfSisasubspaceofavectorspaceV,thenSisavectorspace.第2題isasubspaceof第3題Itispossibletofindapairoftwo-dimensionalsubspacesSandTofsuchthatS∩T={0}.

第4題IfSandTaresubspacesofavectorspaceV,thenS∪TisasubspaceofV.

第5題Ifarelinearlyindependent,thentheyspan.

第6題IfspanavectorspaceV,thentheyarelinearlyindependent.

第7題IfAisanm×nmatrix,thenAandhavethesamerank.

第8題IfAisanm×nmatrix,thenAandhavethesamenullity.

第9題IfAisrowequivalenttoB,thenAandBhavethesamerowspace.

第10題Let

x1,x2,...,xkbelinearlyindependentvectorsin.Ifk<nandxk+1

isavectorthatisnotinSpan

(

x1,x2,...,xk)

,thenthevectors

x1,x2,...,xk,xk+1arelinearlyindependent.

第11題Foreachsetthatfollows,_________isasubspaceof.ABCD第12題GiventhevectorsDeterminewhichofthefollowingoptionsistrue,_______.A,andlinearlyindependentin.Barelinearlyindependent,andtheyspan.Carelinearlyindependentandtheyformabasisfor.Darelinearlyindependentandtheyformabasisfor.第13題Letbeamatrix,andthedeterminantofis0,then_________.ATheremustbeacolumnofelementsinthatareall0.BTheremustbeacolumnofvectorsinthatisalinearcombinationoftheothercolumnvectors.CTheremustbetwocolumnsofelementsproportionalto.DAnycolumnvectorinisalinearcombinationoftheremainingcolumnvectors.第14題Aisanmatrixandisan-dimensionalcolumnvector.Ifthe,thenthelinearequationssystem____________.Amusthaveinfinitemanysolutions.Bmusthavenonzerosolutions.Cmusthaveuniquesolutions.Donlyhavezerosolutions.第15題Let,iftheadjointmatrixofisA?,r(A?)=1.Thentheremusthave________.Aor.Bor.Cand.Dand.Homework4第1題LetbeLetbealineartransformation.If,thenthevectorsandmustbeequal.()第2題If,

arebothlinearoperatorsonavectorspaceV,then

L2?L1isalsoalinearoperatoronV,where

L2?L1

isthemappingde?nedby

（）（L2?L1）(v)=L2(L1(v))forall

v∈V第3題IfIfisalineartransformationandthen()第4題Ifrotateseachvectorxinbydegreesinthecounterclockwisedirectionandthenre?ectstheresultingvectoraboutthey-axis,andifisatransformationthatdoesthesametwooperations,butinthereverseorder,then

L1=L2第5題Thesetofallvectorsxusedinthehomogeneouscoordinatesystemformsasubspaceof

R3第6題Let

bealineartransformation,andletA

bethestandardmatrixrepresentationofL.If

L2isde?nedbyL2(x)=L(L(x))forallx∈R2thenL2isalineartransformationanditsstandardmatrixrepresentationisA2第7題Let

E={x1,x2,...,xn}beanorderedbasisfor

Rn.

If

L1:Rn→Rnand

L2:Rn→RnhavethesamematrixrepresentationwithrespecttoE,then

L1=L2.第8題LetL:Rn→Rnbealineartransformation.If

A

isthestandardmatrixrepresentationofL,

thenan

n×nmatrixB

willalsobeamatrixrepresentationofLifandonlyifB

issimilartoA.

第9題LetA,B

andC

ben×n

matrices.IfA

issimilartoB

andB

issimilartoC,thenA

issimilartoC.第10題Anytwomatriceswiththesametracearesimilar.()第11題Determinewhichthefollowingarenotlinearoperatorson,_________.AListheoperatorde?nedby.BListheoperatorde?nedby.CListheoperatorde?nedby.DListheoperatorde?nedby.第12題Determinewhichthefollowingarenotlinearoperatorson_________.AL(A)=2ABL(A)=ATCL(A)=A+IDL(A)=A?AT第13題LetLbethelinearoperatoronde?nedby

SpanconstitutesthetherangeofLwithdefinedin_________.A.BC,D.第14題Ifthematricesandaresimilar,whichofthefollowingstatementiswrong,__________.AAT

and

BT

aresimilar.BAkand

Bk

aresimilar.CAisrowequivalenttoC，CissimilartoB.Ddet()=det().第15題Thefollowingmatrices,issimilarto,wherethematricesandaredefinedin_______.A.B.C.D.Homewrok5第1題Ifxandyarenonzerovectorsin,thenthevectorprojectionofxontoyisequaltothevectorprojectionofyontox.第2題Ifxandyareunitvectorsinand

|xTy|=1,thenxandyarelinearlyindependent.

第3題IfU,V,andWaresubspacesofandifU⊥VandV⊥W,thenU⊥W.()第4題Itispossibleto?ndanonzerovectoryinthecolumnspaceof

ATsuchthatAy=0.第5題IfAisanm×nmatrix,thenandhavethesamerank.()第6題Ifanm×nmatrixAhaslinearlydependentcolumnsandbisavectorinRm,thenbdoesnothaveauniqueprojectionontothecolumnspaceofA.

第7題IfAisanm×nmatrixsuchthatR()=,thenthesystemAx=bwillhaveauniqueleastsquaressolution.

第8題IfQisanorthogonalmatrix,thenalsoisanorthogonalmatrix.

第9題If{u1,u2,...,uk}isanorthonormalsetofvectorsinandthen(thek×kidentitymatrix).()第10題If{u1,u2,...,uk}isanorthonormalsetofvectorsinandthen(then×nidentitymatrix).

第11題LetxandybevectorsinandletQbeann×northogonalmatrix.Ifz=Qxandw=Qythentherelationshipbetweenanglebetweenzandwandtheanglebetweenxandyis_________.A>B<C=DUncertain第12題LetAbea9×6matrixwithrankequalto5andletbbeavectorin.ThedimensionofN()is______..A9B6C4D5第13題Letandbe

vectorsinaninnerproductspaceV.

ThentherelationofandisA≤B≥C<D>第14題If,thenthescalarproductofxandyis________.A11B8C20D6第15題Theleastsquaressolutiontothesystemis_________.ABCDHomework6第1題IfAisann×nmatrixwhoseeigenvaluesareallnonzero,thenAisnonsingular.()第2題IfAisann×nmatrix,thenAandhavethesameeigenvectors.()第3題IfAandBaresimilarmatrices,thentheyhavethesameeigenvalues.()第4題IfAandBaren×nmatriceswiththesameeigenvalues,thentheyaresimilar.()第5題IfAhaseigenvaluesofmultiplicitygreaterthan1,thenAmustbedefective.()第6題IfAisa5×5matrixofrank1and=0isaneigenvalueofmultiplicity4,thenAisdiagonalizable.()第7題IfAisa6×6matrixofra