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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
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