线性代数难点解析（Analysisofthedifficultiesoflinearalgebra）

线性代数难点解析(Analysisofthedifficultiesoflinear

algebra)

(A+B)=A2+2AB+BA+B2indicates2AB+A2+B2

(AB)=2(AB)(AB)indicatesA2B2

(AB)kindicatesAkBk

(A+B)indicatesA2-B2(A-B)

AlloftheaboveisonlytrueifAandBcanbeexchanged,i.e.,

AB=BA.

3)youcan，tgetA=0orB=0fromAB=0

4)wecan，tgetB=CfromAB=AC

5)A2=Acan'tgetA=IorA=0

6)youcan，tgetA=0fromA2=0

7)thedifferencebetweenmultiplicationandmultiplication

2inversematrix

1)(A-1)-1=A

2)(kA)-1=(1/k)A-1,(kdoesnotequal0)

3)(AB)-1=B-1-1a

4)(A-1)T=(AT)-1

5)||A-1=||-1A

3.Matrixtranspose

1)(AT)T=A

2)(kA)T=kAT,(kisanyrealnumber)

3)(AB)T=BTAT

T=4)(A+B)AT+BT

4.Adjointmatrix

1)A*A=AA*=||I(AB)*=B**A

2)(A*)*=||||n-2A*A=||An-1,(n2or

higher)

3)(kA)*=kn-1A*(A*)T=(AT)*

4)ifr(A)=n,thenr(A*)=n

Ifr(A)=n-1,thenr(A*)=1

Ifr(A)

5)ifAreversible,(A*)-1=(1/A)A,(A*)-1=

(A-1)*,A*=||A-1A

5.Elementarytransformation(3kinds)

1)tworows(column)

2)multiplyalltheelementsink(knot0)timesarow(column)

3)thecorrespondingelementoftheelementofarow(column)

toanotherrow(column)

Note:therank,row,columntransformcanbeusedfortherank

ofprimarytransformation

Toinvertmatrices,onlyroworcolumntransformscanbeused

Thesolutionofalinearsystem,youcanonlyusetherow

transformation

6.Elementarymatrix

1)thematrixobtainedbytheinitialtransformationoftheunit

matrix

(2)firstwaitforPleft(right)timesA,theobtainedPA(AP)

isthesamerow(column)transformationasP

3)theinitialarrayisreversible,anditsinverseisthe

initialmatrixofthesametype

E-lij=Eij,E(-1)I(k)=Ei(1/k),EM)ij(k)=Eij(-k).

7.Matrixequation

1)anequationcontaininganunknownmatrix

2)sufficientandnecessaryconditionsforsolvingmatrix

equations

AXisequaltoBandeachofthecolumnsthathavethesolutions

<==>Bcanberepresentedlinearlybythecolumnvectorsof

A

<==>r(A)=r(A;B)

Iv.Questionsandideas

1.Propositionsabouttheconceptsandpropertiesofmatrices

2.Operationofmatrix(addition,multiplication,

multiplication,transpose)

3.Determinationofmatrixinvertability

Then-thsquarematrixAreversibleisequalto>andthereare

nordersquareB,andAB=BA=I

<==>||Aindicateszero

<==>r(A)=n

<==>Acolumn(row)vectorgroupislinearlyindependent

<===>Ax=0,only0

<==>arbitraryb,soAx=balwayshasauniquesolution

<==>Aisnotzero

4.Inversematrix

1)definemethod:findBtomakeAB=IorBA=I

2)alongwiththelaw:A-1=(1/||A)A*

Note:whenusingthismethod,thealgebraicsubtypeoftherow

shouldbewrittenverticallyinA*.WhencalculatingAij,don，t

omit(-1)I+j.Whenn>3,theprimarytransformationmethod

isusuallyused.

3)methodofelementarytransformation:theline(A;I)only

transforminto(I;A-1)

4)blockmatrixmethod

SolvingmatrixequationAXisequaltoB

1)ifAisinvertible,thenXisequaltoAminusIB,butyou

cansolveforAminus1,andthenmultiplyAminusIBtosolve

forX

2)ifAisinvertible,theinitialtransformationmethodcan

beusedtosolveforXdirectly

(A;B)elementaryrowtransformation(I;X)

IfAisnotinvertible,thentheequationsoftheunknowncolumn

arereducedbygaussianeliminationtothestaircasesystem,

andtheneachcolumnofconstantsisevaluatedseparately.

Thethirdchapterislinearequations

A,key

1,understanding:vector,vectoroperations,andalinear

combinationofthevectorandlineartable,theconceptof

maximumlinearlyindependentgroup,linearcorrelationand

linearlyindependentconcept,theconceptofrankofvector

group,rankofmatrix,theconceptandnatureoftheconcept

ofbasicsolutionsystem.

2.Grasp:theoperationandoperationruleofvectors,the

calculationofmatrixrank,thestructureofsolutionof

homogeneousandnon-homogeneous1inearequations.

3.Application:linearcorrelation,linearlyindependent

judgment,determinationoflinearequations,homogeneousand

non-homogeneous1inearequations.

Second,thedifficulty

Linearlydependent,linearlyindependentdetermination.The

rankoftherankofthevectorgrouptotherankofthematrix.

Therelationshipbetweenthesystemandthelinear

representationofthevectorgroupandtherank.

lii.Analysisofkeydifficulties

Theconceptandoperationofthendimensionalvector

1)concept

2)operation

Ifalphaisalpha(al,a2,…，an)T,beta=(bl,b2,…，bn)

T

Alpha+beta=(al+bl,a2+b2,…an+bn)T

Numberoftimes:kalpha=(kal,ka2,…，kan)T

(alphabeta)=albl+a2b2+…+anbn=alphaTbeta=beta

Talpha

2.Linearcombinationandlineartable

3.Linearcorrelationisindependentoflinearity

1)concept

2)necessaryandnecessaryconditionsforlinearcorrelation

andlinearity

Linearcorrelation

Alpha1,alpha2,...Alphaslinearcorrelation

Alpha1,alpha2,alphas,alphas,xl,x2,xs,T=0have

non-zerosolutions

Therankrofthe>vectorgroupalpha1,alpha2,alphas,alpha

s,thenumberofvectors

<===>hasanalphaI(I=1,2,...s)canberepresented

bytheothers-1vector

Special:nandvectorlinearcorrelation<==>|alpha1

alpha2...Alphan|=0

Nplusonendimensionalvectorsmustbelinearlydependent

Linearlyindependent

Alpha1,alpha2,...Alphasislinearlyindependent

Alpha1,alpha2,alphas,alphas,xl,x2,xs,Tisequalto

zero

Therankrofthe>vectorgroupalpha1,alpha2,alphasis

equaltos(thenumberofvectors)

EachofthevectorsalphaI(I=1,2,…，s)cannotbe

representedbytherestofthes-1vectors

Importantconclusion

A,theechelonvectorgroupmustbelinearlyindependent

B,ifthealpha1,alpha2,...Alphasislinearlyindependent.

Alphai1,alphai2.AlphaItmustbe1inearlyindependent,and

anyofitsextendedgroupsmustbelinearlyindependent.

C,twoorthogonal,non-zerovectorgroupsarelinearly

independent.

4.Rankofvectorgroupandrankofmatrix

1)theconceptofamaximal1inearindependentgroup

2)rankofvectorgroup

3)rankofmatrix

(1)r=r(A)(AT)

(2)r(A+B)orlessr+r(A)(B)

R(kA)=r(A),kdoesnotequal0

(4)r(AB)minorless(r(A),r(B))

IfAisinvertible,thenr(AB)=r(B);

IfBisinvertible,thenr(AB)=r(A)

Andthenwehavethembynmatrix,andBisthenbypmatrix,

ifABisequalto0,thenr(A)+r(B)islessthanorequal

ton

4)therankofthevectorgroupandtherankofthematrix

R(A)=A'srowrank(rankoftherowvectorgroupofmatrix

A)=columnrankofA(rankofcolumnvectorgroupofmatrix

A)

Therankofthevectorgroupisinvariantafterelementary

transformationmatrixandvectorgroup

(3)ifthevectorgroup(I)by(II)lineartable,isr(I)

r(II)orless.Inparticular,theequivalentvectorgrouphas

thesamerank,butthesamevectorgroupoftherankisnot

necessarilyequivalent.

5.Conceptandmethodofbasicsolutions

1)concept

2)methodof

MakeelementaryrowtransformationintoAsteppedonAmatrix,

accordingtothefirstnonzerocoefficientsineachlineofthe

non-zerorepresentedbytheunknownistheprimary(totalof

r(A)Aprincipalcomponent),thenlefttootherunknownisfree

variables(Atotalofn-r(A)A),thefreevariablesaccording

tothestepaftertheassignment,againintosolvingbased

solutionisavailable.

6.Thedeterminationofnon-zerosolutionofhomogeneoussystem

1)let'ssaythatAisthembynmatrix,andAxisequalto

0andthenecessaryconditionsfornon-zerosolutionsarer(A)

<n,whichisAlinearcorrelationofcolumnvectorsofA.

2)ifAisnordermatrix,theAx=0isnotsufficientand

necessaryconditionsforthezerosolutionis||A=0

3)thesufficientconditionforanon-zerosolutiontoAxis

0ism<n,whichisthenumberofequations

Non-homogeneous1inearequationshavethedeterminationof

solutions

1)let'ssaythatAisthembynmatrix,andAxisequalto

bandthenecessaryconditionisthattherankofAisequal

totherankoftheaugmentedmatrix(A),whichisr(A)=r(A).

Let'ssaythatAisthembynmatrix,thesystemAxisequal

tob

Theonlysolutionis<=>r(A)=r(A+)=n

Thereisaninfinitenumberofsolutions,whichareequalto>

r(A)=r(A).

Nosolution<==>r(A)+1=r(A)

Thestructureofthesolutionofnonhomogeneouslinear

equations

SoifwehavenofthelinearequationsAxisequaltob,we

havesolutions,let'ssayeta2,Etatisthefundamental

solutionofthecorrespondinghomogeneoussystem,Ax=0,and

xiisasolutiontoAxisequaltob,kletaoneplusk2eta

twoplus...Plusktetatplusxiisthegeneralsolutionto

Axisequaltob.

1)ifxi1,xi2isasolutiontoAxisequaltob,thenxi1

minusxi2isasolutiontoAxisequalto0

2)ifxiisasolutiontoAxisequaltob,etaisasolution

toAxisequalto0,andxi+ketaisstillasolutiontoAx

isequaltob

3)ifAx=bhasauniquesolution,Ax=0haszerosolutions;

Incontrast,Ax=bhasnoinfinitenumberofsolutionswhen

Ax=0iszero.

Iv.Questionsandideas

1.Propositionsabouttheconceptandnatureofn-dimensional

vectors

2.Additionandmultiplicationofvectors

3.Linearcorrelationandlinearlyindependentproof

1)definition

Aklalpha1+k2alpha2+...+ksalphas=0,andthendo

theidentitytransformationontheupperformula(togetcloser

toknownconditions!)

B=C,AB=AC,soyoucanmultiplybysomeAbytheinformation

oftheknownconditions

Inordertoarrangetheequations,theequationsare

transformedintohomogeneouslinearequationswithknown

conditions.Finally,theanalysisprovesthatkl,k2….，the

valueofks,drawthedesiredconclusion.

2)therank(=thenumberofvectors)

3)thereareonlyzerosolutionstothehomogeneoussystem

4)contradiction

4.Findtherankofthegivenvectorgroupandthemaximallinear

independentgroup

I'mgoingtousetheelementarytransformationmethod,

Thevectorgroupisthematrix,whichissolvedbyelementary

transformation.

5.Therankofmatrix

Commonelementarytransformationmethod.

6.Solvinghomogeneouslinearequationsandnon-homogeneous

linearequations

Chapterivlinearspace

A,key

1.Understanding:conceptsoflinearspace,basis,dimension,

innerproduct,length,Angleanddistance,orthogonalvector

groupandorthonormalbasis,orthogonalmatrix

2.GrasptheoperationrulesofRnanditsvector.

Calculationofinnerproduct,length,Angleanddistance.

3.Application:orthogonaltotwovectors.

Second,thedifficulty

Thepropertiesandapplicationsoforthogonalmatrices.

lii.Analysisofkeydifficulties

1.Conceptsandpropertiesoflinearspaceandbasis

2.Innerproduct,distanceandAngle

1)innerproduct:alphabeta=albl+a2b2+…+anbn

2)length:thesquarerootofalphaalpha(alphaalpha)=(al2

+a22+...+an2)

3)distance:d=equaltothesquarerootofalphabeta(al-

bl)2+(a2-b2)2+...+(an-bn)2)

4)AngleofAngle:costhetatheta=(alphabeta)/(

Thetaisequaltoarccosineofalphabetaofalphabeta.

5)orthogonal:anAngleof90,alphaandbeta,toalphabeta

coming

Alphabetaisorthogonaltobeta

6)orthogonalvectorgroup:anytwovectorsareperpendicular

toeachother

Anygroupofnon-zeroorthogonalvectorsmustbelinearly

independent

ThenumberofvectorsinRnisnotgreaterthann

Theorthogonalizationofthevector

1)theconceptoforthonormalbasis

2)schmidtorthogonalization(normalized,renormalized)

4.Orthogonalmatrix

1)concept

2)thenatureofthe

IfAisorthogonalarray==>=1Aor1

Sothisisequalto>Aminus1isstillorthogonalmatrices

Soit'sgoingtobeequalto>,soit'sgoingtobeequalto

b

==>Aminus1=AT

3)n-ordersquareAisanorthonormalbasisforRnwithnrows

oftheorthogonalarray<==>A

Thencolumnvectorsof==>Aconstituteanorthonormalbasis

forRn

Iv.Questionsandideas

1.Determinewhetheragivensetisalinearspace

Itisgenerallydeterminedbythedefinitionandnatureof

linearspace

Findthebasisanddimensionofthe1inearspace

3.Verifythatthen-dimensionalvectorgroupisanorthonormal

basisforRn

Step:1)theproofvectorisorthogonal,thatis,theinner

productiszero

2)eachvectorisaunitvector,whichis1

4.CalculatetheinnerproductoftwovectorsandtheAngleand

distancebetweenvectors

Thestandardisnormalizedforthegivenvectorgroup

Step:1)determinethelinearcorrelationofvectorgroups,and

onlylinearlyindependentvectorgroupscanbenormalized

2)orthogonalization(schmidtorthogonalization)

3)standardizationvi=betaI/thewholelotbetaI

6.Provethepropositionaboutorthogonalmatrix

Thedecisionoftheorthogonalmatrix

1)definition:ifAAT==>Aisanorthonormalmatrix

IfAATdoesnotequalto==>Aisnotorthogonalarray

Thismethodisusedtoprovetheabstractmatrix.

Then-thsquarematrixAisanorthonormalbasisforRn,which

isannrowvector(orcolumnvector)oftheorthogonalarray

<=>A

Therow(column)oftherow(column)ofAistheunitvector

andit'sorthogonal

Thismethodisusedtogivethematrixofnumericalvalue.

Chapterfiveeigenvaluesandeigenvectors

A,key

1.Understanding:theconceptandbasicpropertiesof

eigenvaluesandeigenvectors.

Theconceptandpropertiesofasimilarmatrixaresimilarto

theconditionsofdiagonalmatrices.

Covenantmatrix.

2.Graspthemethodofcalculatingeigenvalueandeigenvector.

Findasimilardiagonalmatrix.

Second,thedifficulty

Similardiagonalizationanditsapplication.

lii.Analysisofkeydifficulties

1.Theconceptandpropertiesofeigenvaluesandeigenvectors

ofmatrices

1)concept

Note:(1)ifthelambdaisAcharacteristicvalue,then|

lambdaI-A|=0,sothelambdaI-Aisinvertiblematrix

(2)ifthestatusisnotAcharacteristicvalue,thenthe|

IlambdaI-Aindicateszero,sothelambdaI-Aisinvertible

matrix

(3)inparticular,thecharacteristicvalueof0isA<==>

|A=0<==>Airreversible

ThefundamentalsolutiontoAxisequalto0isalinearly

independenteigenvectoroflambdaequalszero

Ifr(A)=1,thenlambda1=sigmaaii,lambda2=lambda3

.Lambda-n—0

2)thenatureofthe

Ifxl,x2arealltheeigenvectorsoftheeigenvaluelambdaI,

thenthe1inearcombinationofxl,x2,andklxlplusk2x2

(non-zero)arestilltheeigenvectorsoflambdaI.The

eigenvectorsoflambdaIarenotunique,andinturn,an

eigenvectorcanonlybelongtooneeigenvalue.

Theeigenvectorsofdifferenteigenvaluesarelinearly

independent,andwhenlambdaIisthekweighteigenvalueof

A,thenumberof1inearlyindependenteigenvectorsoflambda

Iisnotgreaterthank.

Thesumoftheeigenvaluesisequaltothesumoftheelements

inthemaindiagonalofthematrix,andtheproductofthe

eigenvaluesisequaltothevalueofthedeterminantofA.

2.Conceptsandpropertiesofsimilarmatrices

1)concept

2)thenatureofthe

IfA~Bisequalto>AT~BT

==>A-1"b-l(ifAandBarereversible)

Soit'sequalto>Akminusk.

==>|lambdaI-A|=||lambdaI-B,thusAandBhave

thesameeigenvalues

==>||A=B||,thusAandBatthesametime,the

reversibleorirreversible

==>r(A)=r(B)

3.Thematrixcanbesimilartothenecessaryandsufficient

conditionsfordiagonalization

1)theconceptofsimilardiagonalization

2)necessary

Aissimilartothediagonalmatrix,whichhasnlinearly

independenteigenvectors

Ineacheigenvalueof>A,thenumberoflinearlyindependent

eigenvectorsisexactlythesameasthenumberofvaluesofthe

eigenvalue

3)thesufficientconditionforAtobesimilartothediagonal

matrixisthatAhasndifferenteigenvalues

4.Similarityofsymmetricmatrix

1)thesolidsymmetricmatrixmustbediagonalized

2)features

Theeigenvaluesareallreal,andtheeigenvectorsarereal

vectors

Theeigenvectorsofdifferenteigenvaluesareorthogonalto

eachother

Theeigenvaluesofkhavek1inearlyindependenteigenvectors,

orr(lambdaIminusA)=n-k

Iv.Questionsandideas

1.Themethodofeigenvalueandeigenvector

1)theabstractmatrix

Thevaluesofeigenvaluesandeigenvectorsandtheirproperties

derivethevaluesofeigenvalues.

2)thedigitalmatrix

(1)fromthecharacteristicequation|lambdaI-A|=0and

thecharacteristicvalueoflambdaI(n,includingheavyroot)

Thesolutionofthehomogeneoussystem(lambdaIminusA)x=

0,

Thefundamentalsolutionisthelinearlyindependent

eigenvectorsoflambda.

2.DeterminewhetherAcanbediagonalized

1)method1:n-ordersquareAcanbediagonalized<===>

Ahasnlinearlyindependenteigenvectors

Method2:anyeigenvaluelambdaI(settokiweightroot)of

n-ordermatrixAisn-r(lambdail-A)=ki

2)tomakeAAdiagonalmatrixstep

Solet'sfigureouttheeigenvaluesofA,lambda1,lambda2,...,

lambdan

Iwanttofindthecorrespondinglinearindependent

eigenvectorsxl,x2,...,xn

1lambda.

ThereversiblematrixP=(xl,x2...xn),P-1AP=[lambda2...]

Lambdan

3.Takethedeterminantwiththeeigenvalueandthesimilarity

matrix

1)||Alambda=1lambda2...Lambdanwhere:lambda1,lambda

2,...LambdanistheneigenvalueofA

2)ifAandB,then||=||BA

4.UsesimilardiagonalizationforAn

IfA~isAthereisAreversiblematrixP,soP-1AP=the

same

SoAisequaltoP,whichisequaltoPminus1

Amongthem:theequivalentstandardofA

5.Proofofeigenvalueandeigenvector

Chapter6realquadraticform

A,key

1.Understanding:theconceptofquadraticform,the

relationshipbetweenquadraticformandsymmetricarray,the

conceptofmatrixcontract,theconceptofstandardand

normativestandard,theconceptofquadraticandpositive

definitematrix.

Master:thequadraticformandthequadraticform.

Therelationshipbetweencontractsandthetransformationof

thewesternvariables.

Thejudgmentofthequadraticformandthepositivedefinite

matrix.

Application:orthogonaltransformationmethod,matching

methodandelementarytransformationmethodarestandard,the

standardmodelisthestandard.

Second,thedifficulty

Thesecondisstandard.

lii.Analysisofkeydifficulties

1.Theconceptofquadraticformanditsstandardtype

1)quadraticform

Thequadraticmatrixisunique,andthequadraticformshould

beabletowritethesecondmatriximmediately.Conversely,the

realsymmetricmatrixshouldbeabletoconstructthequadratic

form.

2)quadraticstandard

(1)concept

Thepositiveandnegativeinertialindices,r(f)=r(A)=p

+q

WhentheorthogonaltransformationisAstandardtype,the

squarecoefficientofthestandardtypemustbetheneigenvalue

ofthematrixA,andthedistributionmethoddoesnothavethis

property.

3)inertiatheorem

Thepositiveandnegativeinertialindexofquadraticformis

theonlyconstant,whichreflectstheessenceofquadrat