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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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