線性代數(shù)-英文講義

Chapter1MatricesandSystemsofEquationsSystemsofLinearEquationsWheretheaij’s

andbi’sareallrealnumbers,xi’sarevariables.Wewillrefertosystemsoftheform(1)asm×nlinearsystems.DefinitionInconsistent:Alinearsystemhasnosolution.Consistent:Alinearsystemhasatleastonesolution.Example(ⅰ)x1+x2=2x1?x2=2(ⅱ)x1+x2=2x1

+x2=1

(ⅲ)x1+x2=2

?x1

?

x2=-2DefinitionTwosystemsofequationsinvolvingthesamevariablesaresaidtobeequivalentiftheyhavethesamesolutionset.ThreeOperationsthatcanbeusedonasystemtoobtainanequivalentsystem:Ⅰ.Theorderinwhichanytwoequationsarewrittenmaybeinterchanged.Ⅱ.Bothsidesofanequationmaybemultipliedbythesamenonzerorealnumber.Ⅲ.Amultipleofoneequationmaybeaddedto(orsubtractedfrom)another.

n×nSystemsDefinition

Asystemissaidtobeinstricttriangularformifinthekthequationthecoefficientsofthefirstk-1variablesareallzeroandthecoefficientofxkisnonzero(k=1,…,n).isinstricttriangularform.ExampleThesystem

ExampleSolvethesystem

ElementaryRowOperations:Ⅰ.Interchangetworows.Ⅱ.Multiplyarowbyanonzerorealnumber.Ⅲ.Replacearowbyitssumwithamultipleofanotherrow.

ExampleSolvethesystem

2RowEchelonFormpivotalrowpivotalrowDefinition

Amatrixissaidtobeinrowechelonformⅰ.Ifthefirstnonzeroentryineachnonzerorowis1.ⅱ.Ifrowkdoesnotconsistentirelyofzeros,thenumberofleadingzeroentriesinrowk+1isgreaterthanthenumberofleadingzeroentriesinrowk.ⅲ.Iftherearerowswhoseentriesareallzero,theyarebelowtherowshavingnonzeroentries.Example

Determinewhetherthefollowingmatricesareinrowechelonformornot.Definition

TheprocessofusingoperationsⅠ,Ⅱ,ⅢtotransformalinearsystemintoonewhoseaugmentedmatrixisinrowechelonformiscalledGaussianelimination.Definition

Alinearsystemissaidtobeoverdetermined

iftherearemoreequationsthanunknows.Asystemofmlinearequationsinn

unknowsissaidtobeunderdeterminediftherearefewerequationsthanunknows(m<n).ExampleDefinition

Amatrixissaidtobeinreducedrowechelonform

if:ⅰ.Thematrixisinrowechelonform.ⅱ.Thefirstnonzeroentryineachrowistheonlynonzeroentryinitscolumn.HomogeneousSystemsAsystemoflinearequationsissaidtobehomogeneousiftheconstantsontheright-handsideareallzero.Theorem1.2.1

Anm×nhomogeneoussystemoflinearequationshasanontrivialsolutionifn>m.

3MatrixAlgebraMatrixNotationVectorsrowvectorcolumnvector1×nmatrixn×1matrixDefinition

Twom×nmatricesAandBaresaidtobeequalifaij=bijforeachiandj.ScalarMultiplicationIfAisamatrixand

kisascalar,thenkAisthe

matrixformedbymultiplyingeachoftheentriesofAbyk.Definition

IfAisanm×nmatrixandkisascalar,thenkAisthem×n

matrixwhose(i,j)entryiskaij.MatrixAdditionTwomatriceswiththesamedimensionscanbeaddedbyaddingtheircorrespondingentries.

Definition

IfA=(aij)andB=(bij)arebothm×n

matrices,thenthesumA+Bisthem×n

matrixwhose(i,j)entryisaij+bijforeachorderedpair(i,j).ExampleLetThencalculate。MatrixMultiplicationDefinition

IfA=(aij)isanm×nmatrixandB=(bij)

isann×r

matrix,thentheproductAB=C=(cij)isthem×r

matrix

whoseentriesaredefinedby

cij=ai1b1j+ai2b2j+…+ainbnj=

aikbkj.k=1nExamplethencalculateAB.1.If2.IfthencalculateABandBA.MatrixMultiplicationandLinearSystemsCase1OneequationinSeveralUnknowsIfweletandthenwedefinetheproductAXby

Case2

MequationsinN

UnknowsIfweletandthenwedefinetheproductAXby

Definition

Ifa1,a2,…,anarevectorsin

Rmandc1,c2,…,cn

arescalars,thenasumoftheform

c1a1+c2a2+‥‥cnan

issaidtobealinearcombinationofthevectorsa1,a2,…,an

.Theorem1.3.1

(ConsistencyTheoremforLinearSystems)AlinearsystemAX=bisconsistentifandonlyifbcanbewrittenasalinearcombinationofthecolumnvectorsofA.

Theorem1.3.2

EachofthefollowingstatementsisvalidforanyscalarskandlandforanymatricesA,BandCforwhichtheindicatedoperationsaredefined.

A+B=B+A(A+B)+C=A+(B+C)(AB)C=A(BC)

A(B+C)=AB+AC

(A+B)+C=AC+BC(kl)A=k(lA)

k(AB)=(kA)B=A(kB)(k+l)A=kA+lA

k(A+B)=kA+kB

TheIdentityMatrixDefinition

Then×nidentityisthematrixwhereMatrixInversionDefinition

Ann×nmatrixAissaidtobenonsingularorinvertibleifthereexistsamatrixBsuchthatAB=BA=I.ThenmatrixBissaidtobeamultiplicativeinverseofA.Definition

Ann×nmatrixissaidtobesingularifitdoesnothaveamultiplicativeinverse.Theorem1.3.3

IfAandBarenonsingularn×nmatrices,thenABisalsononsingularand(AB)-1=B-1A-1

TheTransposeofaMatrixDefinition

Thetransposeofanm×nmatrixAisthen×mmatrixBdefinedby

bji=aij

forj=1,…,nandi=1,…,m.ThetransposeofAisdenotedbyAT.AlgebraRulesforTranspose:(AT)T=A(kA)T=kAT(A+B)T=AT+BT(AB)T=BTATDefinition

Ann×nmatrixAissaidtobesymmetricifAT=A.4.ElementaryMatricesIfwestartwiththeidentitymatrixIandthenperformexactlyoneelementaryrowoperation,theresultingmatrixiscalledanelementarymatrix.TypeI.AnelementarymatrixoftypeIisamatrixobtainedbyinterchangingtworowsof

I.Example

LetandletAbea3×3matrixthenTypeII.AnelementarymatrixoftypeIIisamatrixobtainedbymultiplyingarowofIbyanonzeroconstant.Example

LetandletAbea3×3matrixthenTypeIII.AnelementarymatrixoftypeIIIisamatrixobtainedfromIbyaddingamultipleofonerowtoanotherrow.Example

LetandletAbea3×3matrixIngeneral,supposethatEisann×nelementarymatrix.Eisobtainedbyeitherarowoperationoracolumnoperation.IfAisann×rmatrix,premultiplying

AbyEhastheeffectofperformingthatsamerowoperationonA.IfB

isanm×nmatrix,postmultiplying

BbyEisequivalenttoperformingthatsamecolumnoperationonB.Example

Let,

Findtheelementarymatrices

，，suchthat.Theorem1.4.1

IfEisanelementarymatrix,thenEisnonsingularandE-1isanelementarymatrixofthesametype.

Definition

AmatrixBisrowequivalenttoAifthereexistsafinitesequenceE1,E2,…,Ekofelementarymatricessuchthat

B=EkEk-1‥‥E1ATheorem1.4.2

(EquivalentConditionsforNonsingularity)LetAbean

n×nmatrix.Thefollowingareequivalent:

Aisnonsingular.

Ax=0hasonlythetrivialsolution0.

AisrowequivalenttoI.

Theorem1.4.3

ThesystemofnlinearequationsinnunknownsAx=bhasauniquesolutionifandonlyifAisnonsingular.

IfAisnonsingular,thenAisrowequivalenttoIandhencethereexistelementarymatricesE1,…,Eksuchthat

EkEk-1‥‥E1A=I

multiplyingbothsidesofthisequationontheright

byA-1EkEk-1‥‥E1I=A-1

Thus(AI)

(IA-1)rowoperationsAmethodforfindingtheinverseofamatrixExampleComputeA-1ifExampleSolvethesystemDiagonalandTriangularMatricesAnn×nmatrixAissaidtobeuppertriangularifaij=0fori>j

andlowertriangularifaij=0fori<j.Ann×nmatrixAissaidtobediagonalifaij=0wheneveri≠j.Aissaidtobetriangularifitiseitheruppertriangularorlowertriangular.5.PartitionedMatricesC=

-2413111132-1246224

C11

C12=

C21

C22

-121B=231141=(b1,b2,b3)AB=A(b1,b2,b3)=(Ab1,Ab2,Ab3)Ingeneral,ifAisanm×nmatrixandBisann×rthathasbeenpartitionedintocolumns(b1,…,br),thentheblockmultiplicationofAtimesBisgivenby

AB=(Ab1,Ab2,…,Abr)IfwepartitionAintorows,then

ThentheproductABcanbepartitionedintorowsasfollows:BlockMultiplicationLetAbeanm×nmatrixandBann×r

matrix.Case1B=(B1

B2),whereB1isann×tmatrixandB2isan