Louis H. Kauffman- Math 310- Gaussian Elimination and Row-Echelon Form

8/3/2019 Louis H. Kauffman- Math 310- Gaussian Elimination and Row-Echelon Form

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Math310SampleProblems

GaussianEliminationandRow-EchelonForm

Problem1:UsetheReducedRow-Echelonformtofindallsolutionsoftheequationsx+3y+z=3

2x+5y+z=83x+8y+2z=11

Youmustshowallyourstepsandworkforcredit.

Problem2:Findthegeneralsolutionof

133269133

x1x2x3

=

155

.

Problem3:(a)Findtherow-reducedechelonformof

A=

123

456789

.

(b)WhatarethesolutionsofthesystemAx=0?(Check!)

Problem4:Giventheequations

x+2y+3z3w=1

4x+5y+6z6w=1

7x+8y+9z

8w=1

a)GivetheReducedRow-Echelonformoftheassociatedaugmentedmatrix.

b)Whicharethefreevariables?Whicharethedependentvariables?

c)Givethegeneralsolutionofthesystemofequations.

Problem5:Giventhetwoequations

x+2y+3z4w=2

2x+4y+3z+w=5

Usethemethodofrowreductiontosolvethesystem.Indicatewhicharethefreevariables,whicharethedependentvariables.Whatisthegeometricinterpretationofthesolution?

Problem6:Leta,b,cbeconstants,andconsiderthesystemofequations

3x+3y+z=ax+y+2z=b5x+5y=c

Findtheequationthattheconstantsa,b,cmustsatisfysothattheseequationsareconsistent.

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MatrixAlgebraandManipulatingMatrices

Problem1:Ineachcase,giveanexampleofamatrixwhichis

nottheidentitymatrix

notthezeromatrix,

andsatisfies:

a)Aisa22diagonalmatrixwithaninverse.

b)Bisa22matrixwithrank1.

c)Cisa22symmetricmatrixwithnoinverse.

d)Oisa22orthogonalmatrix.

So,youmustfindfourmatricesA,B,CandO.

MatrixDeterminants

Problem1:FindthedeterminantofthematrixA=

111124139

.

Problem2:Useeitherthedefinitionofdeterminantintermsofcofactors,orthemethodofrowoperations,tocalculatethedeterminantof

A=

01231111

22331234

Problem3:CalculatethedeterminantofthematrixB=

2001010016201123

.

Problem4:FindthedeterminantofthematrixA3

whereA=

5162

.

Problem5:GiventhematricesA=

5231

,B=

2305

,C=

8654

,

calculatethefollowingdeterminants:

a)|A|,|B|and|C|

b)|ABC2|

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c)|7B|

d)|A7

B|

e)|BC1

|

f)|BT

CA1

|

g)|AB|

Problem6:a)FindthedeterminantofthematrixA=

111124137

.

b)Usethesolutiontoparta)toexplainhowmanysolutionstheequationAx=bhas,where

x=

x

y

z

andb=

000

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MatrixInverses

Problem1:a)Findtheinverse(byanymethod)ofA=

1235

.

b)UsetheabovetoexpressthesolutionsofAx=bintermsoftheconstantsb1andb2.

Problem2:GivetheformulafortheinverseofA=

abcd

.

Problem3:UsethemethodofGaussianEliminationtofindtheinverseforA=

1232343912

.

Problem4:UsethemethodofCofactorstofindtheinverseforA=

121212121

.

Problem5:Findtheinverseofthefollowingmatrices(andcheckyouranswers.)

Donotuseacalculatoryouwillberequiredtoshowallyourworkandcomputations.

a)C=

101111123

b)C=

123014011

c)A=

100210 3

21

d)A=

1200023000340004

Problem6:ForwhatvaluesofthevariabledoesthematrixDbelowhaveaninverse?Explainyouranswer!

D=

33102500+1

Problem7:LetAbeannnmatrix.SupposethatthesystemofequationsAX=0hasauniquesolution.ExplainwhytheinverseA

1hastoexist.

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VectorSpacesandSubspaces

Problem1:ConsiderthesubsetofvectorsinR2

givenby

S={(x,x2)wherexisanyrealnumber}

IsSavectorsubspace?Justifyyouranswercarefully.

Problem2:Istheset

x

x3

wherexR

avectorsubspaceofR2

?Justifyyouranswer.

Problem3:LetVbethespaceofreal-valuedfunctionsofx.ShowthesolutionsetSoftheequation

f(x)=xf(x)

isasubspaceofV.

Problem4:LetVbethespaceofalldifferentiablefunctionsontheline.LetWbethesubsetofallfunctionsfwhicharesolutionsofthedifferentialequationf

+5f=0.ShowthatthesolutionsetWis

asubspaceofV.

Problem5:LetAmn

beamatrixwithmrowsandncolumns.WhatarethefourfundamentalsubspacesassociatedtoA?Givethedefinitionofeachofthefollowing:

Col(A)=thecolumnspaceofA.

Row(A)=therowspaceofA

Null(A)=thenullspaceofA

Conull(A)=conullspaceofA

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LinearIndependence,Spanning,Basis,andDimension

Problem1:FindabasisforthesubspaceVofR3

spannedbythevectors

u1=

201

,u2=

123

,u3=

145

Problem2:InthespaceP3ofpolynomialsofdegree2orless,arethevectors{1+x,1x,1+x+x2

}linearlydependentorindependent?

Problem3:a)ForA=

110213123116

findabasisfortherowspaceandthecolumnspace.

b)IsAx=bsolvableforallb?

Problem4:Forthevectors

w1=

123

w2=

342

andx=

92

5

Isxinthespanof{w1,w2}?Ifso,writexasalinearcombinationof{w1,w2}.

Problem5:Is[1,2,3]T

inthespanof[4,0,5]T

and[6,0,7]T

?

Problem6:a)FindabasisforthesubspaceofR4

spannedbythevectors

v1=

1

21

0

,v2=

2

53

2

,v3=

2

42

0

,v4=

3

85

4

b)Whatisthedimensionofthespanofthevectors{v1,v2,v3,v4}?

Problem7:Dothevectors1+x,1x,x2

spanthespaceP3ofpolynomialsofdegreeatmost2?

Problem8:Findabasisforthesubspaceof22matricesA=

a1,1a1,2a2,1a2,2

satisfyinga1,1+a2,2=0.

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ColumnSpace,RowSpace,NullSpace,ConullSpace

Problem1:Aisanmnmatrix.Let

Col(A)denotethecolumnspaceofA

Row(A)denotetherowspaceofA

Null(A)denotethenullspaceofA

Conull(A)theco-nullspaceofA

Foreachofthefollowingquestions,youranswershouldbeoneoftheabove4spaces.Justifyyouranswerbystatingwhyyouthinkitiscorrect.

a)Thesetofvectorsperpendiculartothecolumnspaceiswhatspace?

b)ThevectorequationAx=bhasasolutionifbbelongstowhatsubspace?

c)Thesetofvectorsperpendiculartotherowspaceiswhatspace?

d)ThevectorequationAx=bhasauniquesolutionifwhatspaceis{0}?

e)Whatnumberdoyougetifyouaddthedimensionsofall4spaces?

Problem2:Giveabasisforthecolumnspace,rowspaceandnullspaceofthematrix

A=

122134712442

Problem3:Findabasisforthenull-spaceofthematrix

A=

1223

24553678

Problem4:a)FindabasisforthecolumnspaceofA=

123222420

.

b)FindabasisfortheperpendicularspaceCol(A)

c)FindabasisforConull(A)

Problem5:LetB=

1211124241036210

FindabasisforthefourfundamentalspacesofB:thecolumnspace,therowspace,thenullspaceandtheco-nullspace(thenullspaceofthetransposeB

T).

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Problem6:Giventhesystemofequations

x+y+z=c1x+2y+2z=c2x+3y+3z=c3

a)Forwhatvaluesofc=

c1c2c3

doesthesystemhaveasolution?

b)Ifthereexistsasolutionforagivenc,howmanyarethere?

c)Findthebasisfortheco-nullspaceofthematrixassociatedtothesystemofequationsabove.

d)Whatistherelationbetweenyouranswerstoparta)andc)?

Problem7:Aisa35matrixandL:R5

R3

isdefinedbyL(v)=Av.SupposethatAhasrank3.

a)WhatisthedimensionofthekernelofL?

b)WhatisthedimensionoftherangeofL?

ExplainyouranswersintermsofhowyouwouldfindbasisofthesespacesifthematrixofAweregiven!

Problem8:LetA=

213042626393

a)GivetheReducedRowEchelonformofthematrixA

b)Findabasisforthenull-spaceofthematrixA

c)FindabasisforthecolumnspaceofthematrixA

d)WhatisthedimensionofthenullspaceN(A)andthecolumnspaceC(A)?

e)AnswerTrueorFalse,andexplainyouranswer:

TheequationAx=bhasasolutionforeveryvectorbR3.

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ChangeofBasisandCoordinates

Problem1:Findthecoordinatesofp=

13

withrespecttothebasisu1=

21

,u2=

12

.

Problem2:Findthenewcoordinates[a,b,c]T

ofthepointx=[7,5,6]T

withrespecttothebasisforR3

givenbythevectors

v1=

20

0

,v2=

11

0

,v3=

21

3

Problem3:GiventhevectorsinR2

u1=

21

,u2=

12

,v1=

10

,v2=

11

a)FindthetransitionmatrixScorrespondingtochangeofbasisfrom{v1,v2}to{u1,u2}.

b)Findthecoordinateexpressionofp=3v1v2withrespecttothebasis{u1,u2}.

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LinearTransformationsandFindingaMatrixRepresentation

Problem1:LetP3bethespaceofpolynomialsofdegree2.ShowthatthemapL:P3P3givenby

L(p(x))=p(x)xp(x)

islinear.(Here,p(x)denotesthefirstderivativeofthepolynomialp(x).)

Problem2:Findthematrix,inthestandardbasisforR3,forthelineartransformation

L

x

y

z

=

2xyz

x2y+zx+3y+2z

.

b)FindthekernelofL

Problem3:DefinethelineartransformationL:P3P3by

L(p(x))=xp

(x)2xp(x)+p(x)

FindthematrixrepresentingLwithrespecttothebasis{1,x,x2

}ofP3.

Problem4:Findthematrixrepresentationforthelineartransformation

L

xy

=

4xyx+4y

.

withrespecttothebasisv1=

31

andv2=

13

.

Problem5:LetVbethespaceoffunctionswithbasis{sin(x),cos(x),sin(2x),cos(2x)}.

DefinethelineartransformationL:VVby

L(f)=f

+f

4f

a)FindthematrixrepresentingLwithrespecttothegivenbasis.

b)FindthekernelofL

Problem6:LetalineartransformationT:R3

R3

bedefinedby

T(v1,v2,v3)=(3v1+2v2+v3,2v1+v2,v2).

Givethematrix(inthestandardbasis)forT.

Problem7:LetVbethevectorspacespannedbythefunctions{ex,e

2x,e

3x},

andletL:VVbethelineartransformationdefinedbyL(f)=f

2f.a)FindthematrixrepresentingLwithrespecttothebasis{ex,e2x,e3x}ofV.

b)FindthekernelofL.

Problem8:DefinethelineartransformationL:R2

R2

byL(v)=AvwhereA=

2113

.

FindthematrixofLwithrespecttothenewbasisv1=

11

andv2=

12

.

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ChangeofBasisforLinearTransformationsandSimilarity

Problem1:ThelineartransformationL:R2

R2

hasmatrixA=

2113

withrespecttothe

standardbasis{e1,e2}ofR2.FindthematrixofLwithrespecttothenewbasis

v1=

11

,v2=

12

Problem2:a)FindthematrixrepresentationAwithrespecttothestandardbasis{e1,e2}ofR2

forthelineartransformation

L

xy

=

4xyx+4y

.

b)FindthematrixrepresentationBofLwithrespecttothebasisv1=

11

andv2=

11

.

Problem3:LetL:R3

R3

bethelineartransformationgivenbyL

xy

z

=

4y+6z2x3yx+2y+z

.

a)FindthematrixrepresentingLwithrespecttothestandardbasis{e1,e2,e3}ofR3.

b)Usetheanswertoparta)tofindthematrixrepresentingLwithrespecttothenewbasis

v1=

21

1

v2=

121

v3=

22

1

Problem4:GiventhevectorsinR2

u1=

21

,u2=

12

,v1=

10

,v2=

11

a)FindthetransitionmatrixScorrespondingtochangeofbasisfrom{v1,v2}to{u1,u2}.

b)ThelineartransformationL:R2

R2

hasamatrixrepresentationA=

1002

withrespecttothe

basis{u1,u2}.FindthematrixrepresentationBofLwithrespecttothebasis{v1,v2}.

Problem5:Forthevectorsv1=

110

,v2=

101

,v3=

011

a)FindthetransitionmatrixScorrespondingtothechangeofbasisfromthestandardbasis{e1,e2,e3}ofR

3tothenewbasis{v1,v2,v3}.

b)LetL:R3

R3

bethelineartransformationdefinedby

L(v1)=v1,L(v2)=2v2,L(v3)=3v3

FindthematrixrepresentingLwithrespecttothestandardbasis{e1,e2,e3}ofR3.

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Problem6:a)Let{v1,v2}beabasisforR2,andletLbealineartransformationofR

2sothat

L(c1v1+c2v2)=(c1+3c2)v1+(2c1+4c2)v2

FindthematrixrepresentingLwithrespecttothebasis{v1,v2}.

b)Supposethatv1=

11

,v2=

11

.FindthematrixrepresentingLwithrespecttothestandard

basisofR2.

Problem7:LetA=

72154

.DefinethelinearmapL:R

2R

2byL(x)=Ax

a)FindthematrixBforthelinearmapLwithrespecttothenewbasisu1=

25

andu2=

13

.

b)Supposethatphascoordinates

31

withrespecttothebasis{u1,u2}.FindL(p)withrespectto

thebasis{u1,u2}.

c)Supposethatp=u2.FindL101

(u2)=L(L(...L(L(u2))...)).

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Eigenvalues,EigenvectorsandEigenspaces

Problem1:FindtheeigenvaluesandcorrespondingeigenvectorsforA=

120320003

.

Problem2:LetA=

111

111111

.TheeigenvaluesofAare1=3,2=0,3=0.a)Findtheeigenvectorsfortheseeigenvalues.

b)NoteAissymmetric.FindanorthogonalmatrixSwithS1

AS=Ddiagonal.

Problem3:FindtheeigenvaluesandeigenvectorsforA=

3423

.

SystemsofDifferentialEquations

Problem1:Giventhedifferentialequationswithinitialconditions

x

=3x+4y;x(0)=1y

=2x3y;y(0)=2

Findthefunctionsx(t)andy(t).

Problem2:a)Givethegeneralsolutionofthedifferentialsystem

y

1=y1+y2y

2=2y1+4y2

b)Givetheparticularsolutionwheny1(0)=3andy2(0)=1.

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FindingPowers,SquareRootsandExponentialsofMatrices

Problem1:ForthematrixA=

1102

a)Find22matricesSandDsuchthatA=SDS1

b)Useyouranswertoparta)tocalculateA5.

Problem2:ForA=

1102

findthematrixe

A.(Youranswershouldbea22matrix.)

Problem3:ForA=

1214

,findthe22matrixe

tA.

Problem4:ForA=

1110

.

a)CalculateA2,A

3,A

4,A

5.

b)FindtheeigenvaluesandeigenvectorsforA.

c)Useyouranswertopartb)tocalculateA10

.

d)Useyouranswertopartb)togetaformulaforAn

whennisapositiveinteger.

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Louis H. Kauffman- Math 310- Gaussian Elimination and Row-Echelon Form

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Transcript of Louis H. Kauffman- Math 310- Gaussian Elimination and Row-Echelon Form