An Introduction To Linear Algebra - Kuttler (2010).pdf

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AnIntroductionToLinearAlgebra
KennethKuttler
July6,2010
2
Contents
1Preliminaries 9
1.1TheNumberLineAndAlgebraOfTheRealNumbers............. 9
1.2Ordered¯elds....................................10
1.3TheComplexNumbers...............................12
1.4Exercises ......................................15
1.5CompletenessofR .................................16
1.6WellOrderingAndArchimedianProperty....................17
1.7DivisionAndNumbers...............................19
1.8SystemsOfEquations...............................22
1.9Exercises ......................................27
1.10F n ..........................................27
1.11AlgebrainF n ....................................27
1.12Exercises ......................................28
1.13TheInnerProductInF n .............................29
1.14Exercises ......................................31
2MatricesAndLinearTransformations 33
2.1Matrices.......................................33
2.1.1The ij th EntryOfAProduct.......................38
2.1.2ACuteApplication............................40
2.1.3PropertiesOfMatrixMultiplication...................42
2.1.4FindingTheInverseOfAMatrix.....................45
2.2Exercises ......................................49
2.3LinearTransformations ..............................51
2.4SubspacesAndSpans ...............................53
2.5AnApplicationToMatrices............................57
2.6MatricesAndCalculus...............................58
2.6.1TheCoriolisAcceleration.........................59
2.6.2TheCoriolisAccelerationOnTheRotatingEarth...........63
2.7Exercises......................................68
3Determinants 73
3.1BasicTechniquesAndProperties.........................73
3.2Exercises ......................................80
3.3TheMathematicalTheoryOfDeterminants...................82
3.3.1TheFunctionsgn..............................82
3.3.2TheDe¯nitionOfTheDeterminant ...................84
3.3.3ASymmetricDe¯nition..........................85
3.3.4BasicPropertiesOfTheDeterminant ..................87
3
4 CONTENTS
3.3.5ExpansionUsingCofactors........................88
3.3.6AFormulaForTheInverse........................90
3.3.7RankOfAMatrix.............................92
3.3.8SummaryOfDeterminants........................94
3.4TheCayleyHamiltonTheorem..........................95
3.5BlockMultiplicationOfMatrices.........................96
3.6Exercises ......................................100
4RowOperations 105
4.1ElementaryMatrices................................105
4.2TheRankOfAMatrix..............................111
4.3TheRowReducedEchelonForm.........................113
4.4RankAndExistenceOfSolutionsToLinearSystems..............116
4.5FredholmAlternative................................117
4.6Exercises ......................................119
5SomeFactorizations 123
5.1 LU Factorization..................................123
5.2FindingAn LU Factorization...........................123
5.3SolvingLinearSystemsUsingAn LU Factorization...............125
5.4The PLU Factorization..............................126
5.5Justi¯cationForTheMultiplierMethod.....................127
5.6ExistenceForThe PLU Factorization......................129
5.7The QR Factorization...............................130
5.8Exercises ......................................133
6LinearProgramming 137
6.1SimpleGeometricConsiderations.........................137
6.2TheSimplexTableau................................138
6.3TheSimplexAlgorithm..............................142
6.3.1Maximums.................................142
6.3.2Minimums..................................144
6.4FindingABasicFeasibleSolution.........................151
6.5Duality.......................................152
6.6Exercises ......................................156
7SpectralTheory 159
7.1EigenvaluesAndEigenvectorsOfAMatrix...................159
7.2SomeApplicationsOfEigenvaluesAndEigenvectors..............167
7.3Exercises ......................................169
7.4Shur'sTheorem...................................176
7.5TraceAndDeterminant..............................184
7.6QuadraticForms..................................185
7.7SecondDerivativeTest...............................186
7.8TheEstimationOfEigenvalues..........................190
7.9AdvancedTheorems................................192
7.10Exercises ......................................195
CONTENTS
5
8VectorSpacesAndFields 203
8.1VectorSpaceAxioms................................203
8.2SubspacesAndBases................................204
8.2.1BasicDe¯nitions..............................204
8.2.2AFundamentalTheorem.........................204
8.2.3TheBasisOfASubspace.........................207
8.3LotsOfFields....................................208
8.3.1 IrreduciblePolynomials..........................208
8.3.2PolynomialsAndFields..........................212
8.3.3TheAlgebraicNumbers..........................217
8.3.4TheLindemannWeierstrassTheoremAndVectorSpaces.......220
8.4Exercises ......................................220
9LinearTransformations 227
9.1MatrixMultiplicationAsALinearTransformation...............227
9.2 L ( V;W )AsAVectorSpace............................227
9.3TheMatrixOfALinearTransformation.....................229
9.3.1SomeGeometricallyDe¯nedLinearTransformations..........236
9.3.2RotationsAboutAGivenVector.....................239
9.3.3TheEulerAngles..............................241
9.4EigenvaluesAndEigenvectorsOfLinearTransformations ...........242
9.5Exercises ......................................244
10LinearTransformationsCanonicalForms 249
10.1ATheoremOfSylvester,DirectSums......................249
10.2DirectSums,BlockDiagonalMatrices......................252
10.3TheJordanCanonicalForm............................255
10.4Exercises ......................................263
10.5TheRationalCanonicalForm...........................267
10.6Uniqueness.....................................274
10.7Exercises ......................................280
11MarkovChainsAndMigrationProcesses 283
11.1RegularMarkovMatrices.............................283
11.2MigrationMatrices.................................287
11.3MarkovChains...................................287
11.4Exercises ......................................292
12InnerProductSpaces 295
12.1GeneralTheory...................................295
12.2TheGrammSchmidtProcess...........................297
12.3RieszRepresentationTheorem..........................300
12.4TheTensorProductOfTwoVectors.......................303
12.5LeastSquares....................................305
12.6FredholmAlternativeAgain............................306
12.7Exercises ......................................306
12.8TheDeterminantAndVolume ..........................311
12.9Exercises ......................................314

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