linear algebra

608 Pages·2017·3.12 MB·English

with Open Texts
A First Course in
LINEAR ALGEBRA
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VERSION 2017 – REVISION A
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A First Course in Linear Algebra
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IlijasFarah,YorkUniversity
KenKuttler,BrighamYoungUniversity
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BruceBauslaugh JenniferMacKenzie
PeterChow TamsynMurnaghan
NathanFriess BogdanSava
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ClaudeLaflamme RyanYee
MarthaLaflamme EhsunZahedi
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a d v a n c i n g l e a r n i n g
A First Course in Linear Algebra
an Open Text
Base TextRevisionHistory
Current Revision: Version2017 — RevisionA
Extensiveedits,additions,andrevisionshavebeencompletedbytheeditorialstaffatLyryxLearning.
Allnewcontent(textandimages)isreleasedunderthesamelicenseasnotedabove.
• Lyryx:Frontmatterhasbeenupdatedincludingcover,copyright,andrevisionpages.
2017A • I.Farah: contributededitsandrevisions,particularlytheproofsinthePropertiesofDeterminantsII:
SomeImportantProofssection
• Lyryx: The text has been updated with the addition of subsections on Resistor Networks and the
MatrixExponentialbasedonoriginalmaterialbyK.Kuttler.
2016B
• Lyryx:Newexample7.35onRandomWalksdeveloped.
• Lyryx: The layout and appearanceof the text has been updated, includingthe title page and newly
2016A
designedbackcover.
• Lyryx: The content was modified and adapted with the addition of new material and several im-
agesthroughout.
2015A
• Lyryx:Additionalexamplesandproofswereaddedtoexistingmaterialthroughout.
• OriginaltextbyK.KuttlerofBrighamYoungUniversity. ThatversionisusedunderCreativeCom-
mons license CC BY (https://creativecommons.org/licenses/by/3.0/) made possible by
2012A
fundingfromTheSaylorFoundation’sOpenTextbookChallenge.SeeElementaryLinearAlgebrafor
moreinformationandtheoriginalversion.

Contents
Contents iii
Preface 1
1 Systems ofEquations 3
1.1 SystemsofEquations,Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 SystemsOfEquations,AlgebraicProcedures . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Elementary Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Gaussian Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.3 UniquenessoftheReduced Row-EchelonForm . . . . . . . . . . . . . . . . . . 25
1.2.4 Rank andHomogeneousSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 28
1.2.5 Balancing Chemical Reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
1.2.6 DimensionlessVariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.2.7 An Applicationto ResistorNetworks . . . . . . . . . . . . . . . . . . . . . . . . 38
2 Matrices 53
2.1 MatrixArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.1.1 AdditionofMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.1.2 Scalar MultiplicationofMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.1.3 MultiplicationofMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
2.1.4 Theijth Entry ofaProduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.1.5 Properties ofMatrixMultiplication . . . . . . . . . . . . . . . . . . . . . . . . . 67
2.1.6 TheTranspose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
2.1.7 TheIdentityand Inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.1.8 FindingtheInverseofa Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
2.1.9 Elementary Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
2.1.10 Moreon MatrixInverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
2.2 LU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
2.2.1 FindingAn LU FactorizationBy Inspection . . . . . . . . . . . . . . . . . . . . . 99
2.2.2 LU Factorization,MultiplierMethod . . . . . . . . . . . . . . . . . . . . . . . . 100
2.2.3 SolvingSystemsusingLU Factorization . . . . . . . . . . . . . . . . . . . . . . . 101
2.2.4 JustificationfortheMultiplierMethod . . . . . . . . . . . . . . . . . . . . . . . . 102
iii
iv CONTENTS
3 Determinants 107
3.1 BasicTechniquesand Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
3.1.1 Cofactors and2 2 Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . 107
×
3.1.2 TheDeterminantofaTriangularMatrix . . . . . . . . . . . . . . . . . . . . . . . 112
3.1.3 Properties ofDeterminantsI: Examples . . . . . . . . . . . . . . . . . . . . . . . 114
3.1.4 Properties ofDeterminantsII: SomeImportantProofs . . . . . . . . . . . . . . . 118
3.1.5 FindingDeterminantsusingRowOperations . . . . . . . . . . . . . . . . . . . . 123
3.2 ApplicationsoftheDeterminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.2.1 A FormulafortheInverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
3.2.2 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
3.2.3 PolynomialInterpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
4 Rn 145
4.1 Vectorsin Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
4.2 Algebrain Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.2.1 AdditionofVectors in Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
4.2.2 Scalar MultiplicationofVectors inRn . . . . . . . . . . . . . . . . . . . . . . . . 150
4.3 GeometricMeaningofVectorAddition . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4.4 LengthofaVector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
4.5 GeometricMeaningofScalar Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . 159
4.6 ParametricLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
4.7 TheDot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
4.7.1 TheDot Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
4.7.2 TheGeometricSignificanceoftheDotProduct . . . . . . . . . . . . . . . . . . . 170
4.7.3 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
4.8 Planes inRn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
4.9 TheCross Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
4.9.1 TheBox Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
4.10 Spanning,LinearIndependenceand Basisin Rn . . . . . . . . . . . . . . . . . . . . . . . 192
4.10.1 Spanning Set ofVectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
4.10.2 Linearly IndependentSet ofVectors . . . . . . . . . . . . . . . . . . . . . . . . . 194
4.10.3 A Short Applicationto Chemistry . . . . . . . . . . . . . . . . . . . . . . . . . . 200
4.10.4 Subspaces and Basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
4.10.5 Row Space, ColumnSpace, and NullSpace ofaMatrix . . . . . . . . . . . . . . . 211
4.11 Orthogonalityand theGram SchmidtProcess . . . . . . . . . . . . . . . . . . . . . . . . 232
4.11.1 Orthogonaland OrthonormalSets . . . . . . . . . . . . . . . . . . . . . . . . . . 233
4.11.2 OrthogonalMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238