ii MATRIX AND POWER SERIES METHODS Mathematics 306 All You Ever Wanted to Know About Matrix Algebra and Infinite Series But Were Afraid To Ask By John W. Lee Department of Mathematics Oregon State University January 2006 Contents Introduction ix I Background and Review 1 1ComplexNumbers 3 1.1Goals................................. 3 1.2Overview............................... 3 1.3TheComplexNumberSystem.................... 4 1.4PropertiesofAbsoluteValuesandComplexConjugates..... 8 1.5TheComplexPlane......................... 8 1.6CirclesintheComplexPlane.................... 12 1.7ComplexFunctions.......................... 14 1.8SuggestedProblems......................... 15 2 Vectors, Lines, and Planes 19 2.1Goals................................. 19 2.2Overview............................... 19 2.3Vectors................................ 19 2.4DotProducts............................. 24 2.5RowandColumnVectors...................... 27 2.6LinesandPlanes........................... 28 2.7SuggestedProblems......................... 30 II Matrix Methods 33 3 Linear Equations 35 3.1Goals................................. 35 3.2Overview............................... 35 3.3ThreeBasicPrinciples........................ 36 3.4SystematicEliminationofUnknowns................ 40 3.5Non-SquareSystems......................... 45 3.6 EfficientEvaluationofDeterminants................ 47 iii iv CONTENTS 3.7ComplexLinearSystems....................... 48 3.8SuggestedProblems......................... 48 4 Matrices and Linear Systems 51 4.1Goals................................. 51 4.2Overview............................... 51 4.3BasicMatrixOperations....................... 51 4.4LinearSystemsinMatrixForm................... 55 4.5TheInverseofaSquareMatrix................... 57 4.6TransposeofaMatrix........................ 60 4.7SuggestedProblems......................... 60 5 Linear Dependence and Independence 65 5.1Goals................................. 65 5.2Overview............................... 65 5.3DependenceandIndependence................... 65 5.4SuggestedProblems......................... 71 6 Matrices and Linear Transformations 73 6.1Goals................................. 73 6.2Overview............................... 73 6.3MatricesasTransformations..................... 73 6.4MatrixofaLinearTransformation................. 76 6.5Projections.............................. 80 6.6SuggestedProblems......................... 83 7 Eigenvalue Problems 87 7.1Goals................................. 87 7.2Overview............................... 87 7.3BasicProperties........................... 88 7.4SymmetricMatrices......................... 94 7.5SuggestedProblems......................... 99 8 Catch Up and Review 103 III Series Methods 107 9 Taylor Polynomial Approximation 109 9.1Goals.................................109 9.2OverviewandaGlanceAhead...................109 9.3TaylorPolynomials..........................110 9.4ErrorinApproximation.......................113 9.5OtherBasePoints..........................115 9.6SuggestedProblems.........................116 CONTENTS v 10 Infinite Series 121 10.1Goals.................................121 10.2Overview...............................121 10.3Convergence,Divergence,andSum.................121 10.4GeometricSeries...........................125 10.5SeriesofFunctions..........................127 10.6SeriesWithComplexTerms.....................128 10.7PowerSeries..............................129 10.8AlgebraicPropertiesofSeries....................129 10.9SuggestedProblems.........................129 11 Taylor Series Representations 133 11.1Goals.................................133 11.2Overview...............................133 11.3TaylorSeries.............................134 11.4 When is f (x) theSumofitsTaylorSeries?............135 11.5OtherBasePoints..........................137 11.6SuggestedProblems.........................138 12 Series With Nonnegative Terms 141 12.1Goals.................................141 12.2Overview...............................141 12.3SerieswithNonnegativeTerms...................142 12.4TheHarmonicSeries.........................143 12.5ImproperIntegrals..........................144 12.6TheIntegralTest...........................145 12.7ApproximatingSums.........................148 12.8SuggestedProblems.........................149 13 Comparison Tests 153 13.1Goals.................................153 13.2Overview...............................153 13.3BasicComparisonTest........................153 13.4LimitComparisonTest........................155 13.5SuggestedProblems.........................156 14 Alternating Series, Absolute Convergence 159 14.1Goals.................................159 14.2Overview...............................159 14.3AlternatingSeries..........................160 14.4AbsoluteConvergence........................162 14.5ConditionalVersusAbsoluteConvergence.............164 14.6SuggestedProblems.........................164 vi CONTENTS 15 Root and Ratio Tests 167 15.1Goals.................................167 15.2Overview...............................167 15.3TheRootTest............................167 15.4TheRatioTest............................169 15.5ApproximationofSums.......................172 15.6SpecialLimits.............................174 15.7SuggestedProblems.........................174 16 Power Series 179 16.1Goals.................................179 16.2Overview...............................179 16.3ConvergenceofPowerSeries.....................180 16.4AlgebraicProperties.........................183 16.5GeneralPowerSeries.........................184 16.6SuggestedProblems.........................185 17 Analytic Properties of Power Series 189 17.1Goals.................................189 17.2Overview...............................189 17.3CalculusandPowerSeries......................190 17.4UniquenessofPowerSeriesRepresentations............191 17.5Applications..............................192 17.6VariationsonaTheme........................195 17.7Term-by-TermOperationsRevisited................195 17.8 Power Series and DifferentialEquations..............197 17.9SuggestedProblems.........................202 18 Power Series and Complex Calculus 207 18.1Goals.................................207 18.2Overview...............................207 18.3 DifferentialCalculusandPowerSeries...............207 18.4EulerIdentities............................210 18.5SuggestedProblems.........................211 19CatchUpandReview 215 IV Supplemental Material 219 A Lab/Recitation Projects 221 A.1GroupProjects............................221 A.2EnrichmentProjects:.........................255 CONTENTS vii BSampleExams 271 B.1SampleMidterms...........................273 B.2SampleFinals.............................285 CSelectedAnswers 307 viii CONTENTS Introduction WELCOME TO MTH 306! ix x Introduction READ THIS INTRODUCTION!!! *********************** DON’T SKIP IT! REALLY! The material for MTH 306 is arranged in 19 lessons. Each lesson, except for the two catch up and review lessons, has the following format: LESSON Topictobecovered GOALS What you are supposed to learn TEXT Main ideas highlighted and discussed SUGGESTED PROBLEMS Homework You should have the calculus text you used for MTH 251-254. That text also includes some topics covered in this course. You may find it helpful to use you calculus text as a complement to the required spiral-bound text for MTH 306. More details follow. READ ON!!!! Introduction xi PREREQUISITES The OSU General Catalog lists MTH 252, Integral Calculus, as the prereq- uisite for MTH 306 because this course adequately prepares you for the infinite series part of MTH 306; however, MTH 252 leaves a small gap in preparation for the matrix and linear algebra part of MTH 306 related to 3-dimensional Carte- sian coordinates and elementary properties of 2-and3-dimensional vectors. If you have taken MTH 254, Vector Calculus I, you are more than adequately prepared for the matrix and linear algebra part of MTH 306. If not, you can make up any gap in your background with a little self-study of 3-dimensional Cartesian coordinates and of the elementary properties of 2-and3-dimensional vectors. Skim Lesson 2 to see the background in vectors you are expected to have. Then do some self-study before Lesson 3 is covered in class. Just go to the OSU Library, check out any calculus book or precalculus book that covers the elements of vectors in 2-and3-dimensions, and study the material that is coveredinLesson2. THINGS TO WATCH FOR AND DO Skim each upcoming lesson before youcometoclass.Readitagain, carefully, afterwards. It will direct your attention to key points. Use the lesson first to help you learn the material and later to review for tests. Suggested problems should be solved shortly after theclasslecture-nottwo or three days after! The problem sets are labeled as “suggested problems” because your instruc- tor may choose to omit some problems, add other problems, or give entirely different problem assignments. Be on your toes! IN THE END IT IS UP TO YOU Mathematics is not a spectator sport! To learn mathematics you must get fully involved. Take notes during the lectures. Think hard about difficult points. Don’t let confusion stand. Get help as soon as difficulties arise. Don’t wait. The Mathematics Learning Center is a valuable resource; use it. So is your instructor. Attend office hours regularly or make an appointment to get help. Keep a notebook with your neatly written reworked lecture notes and solved homework problems. This will help you organize your study, prepare for tests, and provide