Exercise and Solution Manual f
g g g g
or
A First Course in Linear Algebra
g g g g g
Robert A. Beezer
g g
University of Puget Sound
g g g g
Version 3.00 g
Congruent Press g
,RobertgA.gBeezergisgagProfessorgofgMathematicsgatgthegUniversitygofgPugetgSound,gwheregheghasgbeengongthegf
acultygsinceg1984.g HegreceivedgagB.S.gingMathematicsg(withgangEmphasisgingComputergScience)gfromgthegUnive
rsityg ofg Santag Clarag ing 1978,g ag M.S.g ing Statisticsg fromg theg Universityg ofg Illinoisg atg Urbana-
Champaignging 1982g andg ag Ph.D.g ing Mathematicsg fromg theg Universityg ofg Illinoisg atg Urbana-
Champaigng ing 1984.
Ing additiong tog hisg teachingg atg theg Universityg ofg Pugetg Sound,g heg hasg madeg sabbaticalg visitsg tog theg University
gofgthegWestgIndiesg(Trinidadgcampus)g andgthegUniversityg ofgWesterngAustralia.g Heghasgalsoggivengseveralgcour
sesgingthegMaster’sgprogramgatgthegAfricangInstitutegforgMathematicalgSciences,gSouthgAfrica.g Heghasg beengagS
agegdevelopergsinceg2008.
Hegteachesgcalculus,glineargalgebragandgabstractgalgebragregularly,gwhileghisgresearchginterestsgincludegthegapp
licationsg ofg linearg algebrag tog graphg theory.g Hisg professionalg websiteg isg atg http://buzzard.ups.edu.
Edition
Versiong 3.00
Decemberg 7,g 2012
Coverg Design
Aidang Meacham
Publisherg Rober
tgA.gBeezergCong
ruentgPress
GiggHarbor,gWashington,gUSA
⃝cg 2004—2012g g Robertg A.g Beezer
Permissiongisggrantedgtogcopy,gdistributegand/orgmodifygthisgdocumentgundergthegtermsgofgthegGNUgFreegDoc
umentationg License,g Versiong 1.2g org anyg laterg versiong publishedg byg theg Freeg Softwareg Foundation;g withgnog
Invariantg Sections,g nog Front-Coverg Texts,g andg nog Back-
Coverg Texts.g Ag copyg ofg theg licenseg isg includedg ingthegappendixgentitledg“GNUgFreegDocumentationgLicense”.
Theg mostg recentg versiong cang alwaysg beg foundg atg http://linear.pugetsound.edu.
,Contents
Systemsg ofg Linearg Equations 1
Whatg isg Linearg Algebra?........................................................................................................................................ 1
Solvingg Systemsg ofg Linearg Equations ................................................................................................................... 1
Reducedg Row-Echelong Form ................................................................................................................................. 6
Typesg ofg Solutiong Sets ......................................................................................................................................... 13
Homogeneousg Systemsg ofg Equations ................................................................................................................... 18
Nonsingularg Matrices ........................................................................................................................................... 23
Vectors 28
Vectorg Operations ................................................................................................................................................. 28
Linearg Combinations ............................................................................................................................................. 32
Spanningg Sets ....................................................................................................................................................... 33
Linearg Independence ............................................................................................................................................ 41
Linearg Dependenceg andg Spans............................................................................................................................ 48
Orthogonality ......................................................................................................................................................... 51
Matrices 53
Matrixg Operations ................................................................................................................................................ 53
Matrixg Multiplication ............................................................................................................................................ 57
Matrixg Inversesg andg Systemsg ofg Linearg Equations ........................................................................................... 61
Matrixg Inversesg andg Nonsingularg Matrices ....................................................................................................... 65
Columng andg Rowg Spaces..................................................................................................................................... 67
Fourg Subsets.......................................................................................................................................................... 72
Vectorg Spaces 77
Vectorg Spaces........................................................................................................................................................ 77
Subspaces............................................................................................................................................................... 80
Linearg Independenceg andg Spanningg Sets........................................................................................................... 84
Bases ...................................................................................................................................................................... 91
Dimension .............................................................................................................................................................. 95
Propertiesg ofg Dimension ...................................................................................................................................... 99
Determinants 101
Determinantg ofg ag Matrix .................................................................................................................................... 101
Propertiesg ofg Determinantsg ofg Matrices .......................................................................................................... 104
Eigenvalues 106
Eigenvaluesg andg Eigenvectors ............................................................................................................................ 106
Propertiesg ofg Eigenvaluesg andg Eigenvectors .................................................................................................... 111
Similarityg andg Diagonalization ........................................................................................................................... 113
Linearg Transformations 117
Linearg Transformations....................................................................................................................................... 117
Injectiveg Linearg Transformations....................................................................................................................... 121
Surjectiveg Linearg Transformations .................................................................................................................... 126
Invertibleg Linearg Transformations ..................................................................................................................... 131
iii
, Representations 136
Vectorg Representations...................................................................................................................................... 136
Matrixg Representations ..................................................................................................................................... 137
Changeg ofg Basis .................................................................................................................................................. 146
Orthonormalg Diagonalization ............................................................................................................................ 149
Archetypes 150
iv
g g g g
or
A First Course in Linear Algebra
g g g g g
Robert A. Beezer
g g
University of Puget Sound
g g g g
Version 3.00 g
Congruent Press g
,RobertgA.gBeezergisgagProfessorgofgMathematicsgatgthegUniversitygofgPugetgSound,gwheregheghasgbeengongthegf
acultygsinceg1984.g HegreceivedgagB.S.gingMathematicsg(withgangEmphasisgingComputergScience)gfromgthegUnive
rsityg ofg Santag Clarag ing 1978,g ag M.S.g ing Statisticsg fromg theg Universityg ofg Illinoisg atg Urbana-
Champaignging 1982g andg ag Ph.D.g ing Mathematicsg fromg theg Universityg ofg Illinoisg atg Urbana-
Champaigng ing 1984.
Ing additiong tog hisg teachingg atg theg Universityg ofg Pugetg Sound,g heg hasg madeg sabbaticalg visitsg tog theg University
gofgthegWestgIndiesg(Trinidadgcampus)g andgthegUniversityg ofgWesterngAustralia.g Heghasgalsoggivengseveralgcour
sesgingthegMaster’sgprogramgatgthegAfricangInstitutegforgMathematicalgSciences,gSouthgAfrica.g Heghasg beengagS
agegdevelopergsinceg2008.
Hegteachesgcalculus,glineargalgebragandgabstractgalgebragregularly,gwhileghisgresearchginterestsgincludegthegapp
licationsg ofg linearg algebrag tog graphg theory.g Hisg professionalg websiteg isg atg http://buzzard.ups.edu.
Edition
Versiong 3.00
Decemberg 7,g 2012
Coverg Design
Aidang Meacham
Publisherg Rober
tgA.gBeezergCong
ruentgPress
GiggHarbor,gWashington,gUSA
⃝cg 2004—2012g g Robertg A.g Beezer
Permissiongisggrantedgtogcopy,gdistributegand/orgmodifygthisgdocumentgundergthegtermsgofgthegGNUgFreegDoc
umentationg License,g Versiong 1.2g org anyg laterg versiong publishedg byg theg Freeg Softwareg Foundation;g withgnog
Invariantg Sections,g nog Front-Coverg Texts,g andg nog Back-
Coverg Texts.g Ag copyg ofg theg licenseg isg includedg ingthegappendixgentitledg“GNUgFreegDocumentationgLicense”.
Theg mostg recentg versiong cang alwaysg beg foundg atg http://linear.pugetsound.edu.
,Contents
Systemsg ofg Linearg Equations 1
Whatg isg Linearg Algebra?........................................................................................................................................ 1
Solvingg Systemsg ofg Linearg Equations ................................................................................................................... 1
Reducedg Row-Echelong Form ................................................................................................................................. 6
Typesg ofg Solutiong Sets ......................................................................................................................................... 13
Homogeneousg Systemsg ofg Equations ................................................................................................................... 18
Nonsingularg Matrices ........................................................................................................................................... 23
Vectors 28
Vectorg Operations ................................................................................................................................................. 28
Linearg Combinations ............................................................................................................................................. 32
Spanningg Sets ....................................................................................................................................................... 33
Linearg Independence ............................................................................................................................................ 41
Linearg Dependenceg andg Spans............................................................................................................................ 48
Orthogonality ......................................................................................................................................................... 51
Matrices 53
Matrixg Operations ................................................................................................................................................ 53
Matrixg Multiplication ............................................................................................................................................ 57
Matrixg Inversesg andg Systemsg ofg Linearg Equations ........................................................................................... 61
Matrixg Inversesg andg Nonsingularg Matrices ....................................................................................................... 65
Columng andg Rowg Spaces..................................................................................................................................... 67
Fourg Subsets.......................................................................................................................................................... 72
Vectorg Spaces 77
Vectorg Spaces........................................................................................................................................................ 77
Subspaces............................................................................................................................................................... 80
Linearg Independenceg andg Spanningg Sets........................................................................................................... 84
Bases ...................................................................................................................................................................... 91
Dimension .............................................................................................................................................................. 95
Propertiesg ofg Dimension ...................................................................................................................................... 99
Determinants 101
Determinantg ofg ag Matrix .................................................................................................................................... 101
Propertiesg ofg Determinantsg ofg Matrices .......................................................................................................... 104
Eigenvalues 106
Eigenvaluesg andg Eigenvectors ............................................................................................................................ 106
Propertiesg ofg Eigenvaluesg andg Eigenvectors .................................................................................................... 111
Similarityg andg Diagonalization ........................................................................................................................... 113
Linearg Transformations 117
Linearg Transformations....................................................................................................................................... 117
Injectiveg Linearg Transformations....................................................................................................................... 121
Surjectiveg Linearg Transformations .................................................................................................................... 126
Invertibleg Linearg Transformations ..................................................................................................................... 131
iii
, Representations 136
Vectorg Representations...................................................................................................................................... 136
Matrixg Representations ..................................................................................................................................... 137
Changeg ofg Basis .................................................................................................................................................. 146
Orthonormalg Diagonalization ............................................................................................................................ 149
Archetypes 150
iv
