Lecture notes on Advanced linear Algebra

how linear algebra is used in machine learning and lecture notes on probability statistics and linear algebra. linear algebra and differential equations lecture notes

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Published Date:15-07-2017

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with Open Texts AFirstCoursein LINEARALGEBRA anOpenText BASETEXTBOOK VERSION2017–REVISIONA ADAPTABLEACCESSIBLEAFFORDABLE byLyryxLearning basedontheoriginaltextbyK.Kuttler CreativeCommonsLicense(CCBY)advancing learning ChampionsofAccesstoKnowledge ONLINE OPENTEXT ASSESSMENT All digital forms of access to our high-quality We have been developing superior online for- open texts are entirely FREE All content is mativeassessmentformorethan15years. Our reviewed for excellence and is wholly adapt- questions are continuously adapted with the able; custom editions are produced by Lyryx content and reviewed for quality and sound for those adopting Lyryx assessment. Access pedagogy. To enhance learning, students re- to theoriginal source ﬁles is also open to any- ceive immediate personalized feedback. Stu- one dent grade reports and performance statistics arealsoprovided. 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CONTRIBUTIONS Ilijas Farah,YorkUniversity KenKuttler, Brigham YoungUniversity LyryxLearningTeam Bruce Bauslaugh Jennifer MacKenzie Peter Chow TamsynMurnaghan NathanFriess BogdanSava Stephanie Keyowski Larissa Stone Claude Laﬂamme RyanYee Martha Laﬂamme EhsunZahedi LICENSE CreativeCommonsLicense(CCBY):Thistext,includingtheartandillustrations,areavailableunder theCreativeCommonslicense(CCBY),allowinganyonetoreuse,revise,remixandredistributethetext. Toviewacopyofthislicense,visithttps://creativecommons.org/licenses/by/4.0/advancing learning AFirstCoursein LinearAlgebra anOpenText BaseTextRevisionHistory CurrentRevision: Version2017 — RevisionA Extensiveedits,additions,andrevisionshavebeencompletedbytheeditorialstaffatLyryxLearning. Allnewcontent(textandimages)isreleasedunderthesamelicenseasnotedabove. • Lyryx: Frontmatterhasbeenupdatedincludingcover,copyright,andrevisionpages. 2017A • I. Farah: contributededits and revisions, particularlythe proofsin the Properties of Determinants II: 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 appearance of the text has been updated, including the 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. Thatversionis usedunderCreativeCom- 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 SystemsofEquations 3 1.1 SystemsofEquations,Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 SystemsOfEquations,AlgebraicProcedures . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 ElementaryOperations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2.2 GaussianElimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.2.3 UniquenessoftheReducedRow-EchelonForm . . . . . . . . . . . . . . . . . . 25 1.2.4 Rank andHomogeneousSystems . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.2.5 BalancingChemicalReactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.2.6 DimensionlessVariables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.2.7 AnApplicationtoResistorNetworks . . . . . . . . . . . . . . . . . . . . . . . . 38 2 Matrices 53 2.1 MatrixArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.1.1 AdditionofMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.1.2 ScalarMultiplicationofMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . 57 2.1.3 MultiplicationofMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 th 2.1.4 Theij EntryofaProduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 2.1.5 PropertiesofMatrixMultiplication . . . . . . . . . . . . . . . . . . . . . . . . . 67 2.1.6 TheTranspose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.1.7 TheIdentityandInverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.1.8 FindingtheInverseofaMatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.1.9 ElementaryMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.1.10 MoreonMatrixInverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.2 LU Factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 2.2.1 FindingAnLU FactorizationByInspection . . . . . . . . . . . . . . . . . . . . . 99 2.2.2 LU Factorization,MultiplierMethod . . . . . . . . . . . . . . . . . . . . . . . . 100 2.2.3 SolvingSystemsusingLU Factorization . . . . . . . . . . . . . . . . . . . . . . . 101 2.2.4 JustiﬁcationfortheMultiplierMethod . . . . . . . . . . . . . . . . . . . . . . . . 102 iiiiv CONTENTS 3 Determinants 107 3.1 BasicTechniquesandProperties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.1.1 Cofactors and2×2Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . 107 3.1.2 TheDeterminantofaTriangularMatrix . . . . . . . . . . . . . . . . . . . . . . . 112 3.1.3 PropertiesofDeterminantsI:Examples . . . . . . . . . . . . . . . . . . . . . . . 114 3.1.4 PropertiesofDeterminantsII:SomeImportantProofs . . . . . . . . . . . . . . . 118 3.1.5 FindingDeterminantsusingRowOperations . . . . . . . . . . . . . . . . . . . . 123 3.2 ApplicationsoftheDeterminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.2.1 AFormulafortheInverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 3.2.2 Cramer’s Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 3.2.3 PolynomialInterpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 n 4 R 145 n 4.1 VectorsinR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 n 4.2 AlgebrainR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 n 4.2.1 AdditionofVectorsinR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 n 4.2.2 ScalarMultiplicationofVectorsinR . . . . . . . . . . . . . . . . . . . . . . . . 150 4.3 GeometricMeaningofVectorAddition . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 4.4 LengthofaVector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.5 GeometricMeaningofScalarMultiplication . . . . . . . . . . . . . . . . . . . . . . . . . 159 4.6 ParametricLines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 4.7 TheDotProduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 4.7.1 TheDotProduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 4.7.2 TheGeometricSigniﬁcanceoftheDotProduct . . . . . . . . . . . . . . . . . . . 170 4.7.3 Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 n 4.8 PlanesinR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 4.9 TheCrossProduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 4.9.1 TheBoxProduct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 n 4.10 Spanning,LinearIndependenceandBasisinR . . . . . . . . . . . . . . . . . . . . . . . 192 4.10.1 SpanningSetofVectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 4.10.2 LinearlyIndependentSetofVectors . . . . . . . . . . . . . . . . . . . . . . . . . 194 4.10.3 AShortApplicationtoChemistry . . . . . . . . . . . . . . . . . . . . . . . . . . 200 4.10.4 SubspacesandBasis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 4.10.5 RowSpace, ColumnSpace,andNullSpaceofaMatrix . . . . . . . . . . . . . . . 211 4.11 OrthogonalityandtheGramSchmidtProcess . . . . . . . . . . . . . . . . . . . . . . . . 232 4.11.1 OrthogonalandOrthonormalSets . . . . . . . . . . . . . . . . . . . . . . . . . . 233 4.11.2 OrthogonalMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238CONTENTS v 4.11.3 Gram-SchmidtProcess . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 4.11.4 OrthogonalProjections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 4.11.5 LeastSquaresApproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 4.12 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 4.12.1 VectorsandPhysics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 4.12.2 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 5 LinearTransformations 269 5.1 LinearTransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 5.2 TheMatrixofaLinearTransformationI . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 5.3 PropertiesofLinearTransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 2 5.4 SpecialLinearTransformationsinR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 5.5 OnetoOneandOntoTransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 5.6 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 5.7 TheKernelAndImageOfALinearMap. . . . . . . . . . . . . . . . . . . . . . . . . . . 310 5.8 TheMatrixofaLinearTransformationII . . . . . . . . . . . . . . . . . . . . . . . . . . 315 5.9 TheGeneral SolutionofaLinearSystem. . . . . . . . . . . . . . . . . . . . . . . . . . . 321 6 ComplexNumbers 329 6.1 ComplexNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329 6.2 PolarForm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336 6.3 RootsofComplexNumbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 339 6.4 TheQuadraticFormula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343 7 SpectralTheory 347 7.1 EigenvaluesandEigenvectorsofaMatrix . . . . . . . . . . . . . . . . . . . . . . . . . . 347 7.1.1 DeﬁnitionofEigenvectorsandEigenvalues . . . . . . . . . . . . . . . . . . . . . 347 7.1.2 FindingEigenvectorsandEigenvalues . . . . . . . . . . . . . . . . . . . . . . . . 350 7.1.3 EigenvaluesandEigenvectorsforSpecialTypesofMatrices . . . . . . . . . . . . 356 7.2 Diagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 7.2.1 SimilarityandDiagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 7.2.2 DiagonalizingaMatrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 7.2.3 ComplexEigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 7.3 ApplicationsofSpectralTheory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 7.3.1 RaisingaMatrixtoaHighPower . . . . . . . . . . . . . . . . . . . . . . . . . . 373 7.3.2 RaisingaSymmetricMatrixtoaHighPower . . . . . . . . . . . . . . . . . . . . 375 7.3.3 MarkovMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 7.3.3.1 EigenvaluesofMarkovMatrices . . . . . . . . . . . . . . . . . . . . . 384vi CONTENTS 7.3.4 DynamicalSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384 7.3.5 TheMatrixExponential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 7.4 Orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 7.4.1 OrthogonalDiagonalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 7.4.2 TheSingularValueDecomposition . . . . . . . . . . . . . . . . . . . . . . . . . 409 7.4.3 PositiveDeﬁniteMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 7.4.3.1 TheCholeskyFactorization . . . . . . . . . . . . . . . . . . . . . . . . 420 7.4.4 QRFactorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422 7.4.4.1 TheQRFactorizationandEigenvalues . . . . . . . . . . . . . . . . . . 424 7.4.4.2 PowerMethods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 7.4.5 QuadraticForms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427 8 SomeCurvilinearCoordinateSystems 439 8.1 PolarCoordinatesandPolarGraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 8.2 SphericalandCylindricalCoordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449 9 VectorSpaces 455 9.1 AlgebraicConsiderations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 9.2 SpanningSets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471 9.3 LinearIndependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475 9.4 SubspacesandBasis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483 9.5 SumsandIntersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 498 9.6 LinearTransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499 9.7 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505 9.7.1 OnetoOneandOntoTransformations . . . . . . . . . . . . . . . . . . . . . . . . 505 9.7.2 Isomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509 9.8 TheKernelAndImageOfALinearMap. . . . . . . . . . . . . . . . . . . . . . . . . . . 518 9.9 TheMatrixofaLinearTransformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524 A SomePrerequisiteTopics 537 A.1 SetsandSet Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537 A.2 WellOrderingandInduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539 B SelectedExerciseAnswers 543 Index 591Preface A First Course in Linear Algebra presents an introduction to the fascinating subject of linear algebra for students who have a reasonable understanding of basic algebra. Major topics of linear algebra are pre- sentedindetail,withproofsofimportanttheoremsprovided. Separatesectionsmaybeincludedinwhich proofs are examined in further depth and in general these can be excluded without loss of contrinuity. Where possible, applications of key concepts are explored. In an effort to assist those students who are interestedincontinuingonin linearalgebraconnectionstoadditionaltopicscoveredinadvancedcourses areintroduced. Each chapter begins with a list of desired outcomes which a student should be able to achieve upon completing the chapter. Throughout the text, examples and diagrams are given to reinforce ideas and provide guidance on how to approach various problems. Students are encouraged to work through the suggestedexercises providedat theend ofeach section. Selected solutionsto theseexercisesare givenat theendofthetext. As thisis an open text, youare encouraged to interact withthetextbookthroughannotating, revising, andreusingtoyouradvantage. 11.SystemsofEquations 1.1SystemsofEquations,Geometry Outcomes A. Relate the types of solution sets of a system of two (three) variables to the intersections of linesinaplane(theintersectionofplanesinthreespace) Asyoumayremember,linearequationslike2x+3y=6canbegraphedasstraightlinesinthecoordi- nateplane. We say thatthisequation isin twovariables, in thiscase x and y. Supposeyou havetwosuch equations, each of which can be graphed as a straight line, and consider the resulting graph of two lines. Whatwoulditmeanifthereexistsapointofintersectionbetweenthetwolines? Thispoint,whichlieson both graphs, gives x and y values for which both equations are true. In other words, this point gives the orderedpair(x,y)thatsatisfybothequations. Ifthepoint(x,y)isapointofintersection,wesaythat (x,y) is a solution to the two equations. In linear algebra, we often are concerned with ﬁnding the solution(s) to a system of equations, if such solutionsexist. First, we consider graphical representations of solutions andlaterwewillconsiderthealgebraicmethodsforﬁndingsolutions. When looking for the intersection of two lines in a graph, several situations may arise. The follow- ing picture demonstrates the possible situations when considering two equations (two lines in the graph) involvingtwovariables. y y y x x x OneSolution NoSolutions InﬁnitelyManySolutions Intheﬁrstdiagram,thereisauniquepointofintersection,whichmeansthatthereisonlyone(unique) solutiontothetwoequations. In thesecond, thereareno pointsofintersectionand nosolution. Whenno solutionexists,thismeansthatthetwolinesareparallelandtheyneverintersect. Thethirdsituationwhich can occur, as demonstrated in diagram three, is that the two lines are really the same line. For example, x+y =1 and 2x+2y =2 are equations which when graphed yield the same line. In this case there are inﬁnitely many points which are solutions of these two equations, as every ordered pair which is on the graphofthelinesatisﬁesbothequations. Whenconsideringlinearsystemsofequations,therearealways threetypesofsolutionspossible;exactlyone(unique)solution,inﬁnitelymanysolutions,ornosolution. 34 SystemsofEquations Example1.1:AGraphicalSolution Useagraphtoﬁndthesolutiontothefollowingsystemofequations x+y=3 y−x=5 Solution.Throughgraphingtheaboveequationsandidentifyingthepointofintersection,wecanﬁndthe solution(s). Remember that we must have either one solution, inﬁnitely many, or no solutionsat all. The followinggraphshowsthetwoequations,aswellastheintersection. Remember,thepointofintersection represents the solution of the two equations, or the (x,y) which satisfy both equations. In this case, there isonepointofintersectionat (−1,4)whichmeanswehaveoneuniquesolution,x=−1,y=4. y 6 4 (x,y)=(−1,4) 2 x −4 −3 −2 −1 1 ♠ In the above example, we investigated the intersection point of two equations in two variables, x and y. Nowwewillconsiderthegraphicalsolutionsofthreeequationsintwovariables. Consider a system of three equations in two variables. Again, these equations can be graphed as straightlinesintheplane,sothattheresultinggraphcontainsthreestraightlines. Recallthethreepossible types ofsolutions;no solution,one solution,and inﬁnitelymanysolutions. There are nowmorecomplex ways of achieving these situations, due to the presence of the third line. For example, you can imagine thecaseofthreeintersectinglineshavingnocommonpointofintersection. Perhapsyoucanalsoimagine threeintersectinglineswhichdointersectatasinglepoint. Thesetwosituationsareillustratedbelow. y y x x NoSolution OneSolution1.1.SystemsofEquations,Geometry 5 Consider the ﬁrst picture above. While all three lines intersect with one another, there is no common point of intersection where all three lines meet at one point. Hence, there is no solution to the three equations. Remember, a solution is a point (x,y) which satisﬁes all three equations. In the case of the second picture, the lines intersect at a common point. This means that there is one solution to the three equationswhosegraphsarethegivenlines. Youshouldtakeamomentnowtodrawthegraphofasystem which resultsinthreeparallel lines. Next,try thegraphofthree identicallines. Which typeofsolutionis representedineach ofthesegraphs? Wehavenowconsidered thegraphical solutionsofsystemsoftwoequationsin twovariables,aswell as three equations in two variables. However, there is no reason to limit our investigationto equations in twovariables. Wewillnowconsiderequationsinthreevariables. You may recall that equations in three variables, such as 2x+4y−5z =8, form a plane. Above, we werelookingforintersectionsoflinesinordertoidentifyanypossiblesolutions. Whengraphicallysolving systems of equations in three variables, we look for intersections of planes. These points of intersection give the (x,y,z) that satisfy all the equations in the system. What types of solutions are possible when working with three variables? Consider the following picture involving two planes, which are given by twoequationsinthreevariables. Notice how thesetwo planes intersect in a line. This means that the points (x,y,z) on this line satisfy both equations in the system. Since the line contains inﬁnitely many points, this system has inﬁnitely manysolutions. It could also happen that the two planes fail to intersect. However, is it possible to have two planes intersect at asinglepoint? Take amomenttoattempt drawingthissituation,and convinceyourselfthat it is not possible This means that when we have only two equations in three variables, there is no way to havea uniquesolution Hence, thetypes of solutionspossiblefor two equationsin three variables are no solutionorinﬁnitelymanysolutions. Now imagine adding a third plane. In other words, consider three equations in three variables. What typesofsolutionsarenowpossible? Considerthefollowingdiagram. NewPlane ✠ In this diagram, there is no point which lies in all three planes. There is no intersection between all6 SystemsofEquations planes so there is no solution. The picture illustrates the situation in which the line of intersection of the new plane with one of the original planes forms a line parallel to the line of intersection of the ﬁrst two planes. However, in three dimensions, it is possiblefor two lines to fail to intersect even though they are notparallel. Suchlinesarecalledskewlines. Recall that when working with two equations in three variables, it was not possible to have a unique solution. Is it possiblewhen consideringthree equationsin three variables? In fact, it is possible,and we demonstratethissituationinthefollowingpicture. NewPlane ✠ In this case, the three planes have a single point of intersection. Can you think of other types of solutions possible? Another is that the three planes could intersect in a line, resulting in inﬁnitely many solutions,asinthefollowingdiagram. We have now seen how three equations in three variables can have no solution, a unique solution, or intersect in a line resulting in inﬁnitely many solutions. It is also possible that the three equations graph thesameplane,whichalsoleadstoinﬁnitelymanysolutions. Youcanseethatwhenworkingwithequationsinthreevariables,therearemanymorewaystoachieve thedifferenttypesofsolutionsthanwhenworkingwithtwovariables. Itmayproveenlighteningtospend timeimagining(anddrawing)manypossiblescenarios,andyoushouldtakesometimetotryafew. Youshouldalsotakesometimetoimagine(anddraw)graphsofsystemsinmorethanthreevariables. Equationslikex+y−2z+4w=8withmorethanthreevariablesareoftencalledhyper-planes. Youmay soon realize that it is tricky to draw the graphs of hyper-planes Through the tools of linear algebra, we canalgebraicallyexaminethesetypesofsystemswhicharedifﬁculttograph. Inthefollowingsection,we willconsiderthesealgebraictools.1.2.SystemsOfEquations,AlgebraicProcedures 7 Exercises Exercise 1.1.1 Graphically, ﬁnd the point (x ,y ) which lies on both lines, x+3y =1 and 4x−y = 3. 1 1 Thatis,grapheachlineandsee wheretheyintersect. Exercise1.1.2 Graphically,ﬁndthepointofintersectionofthetwolines3x+y=3andx+2y=1.That is,grapheachlineandseewhere theyintersect. Exercise 1.1.3 You have a system of k equations in two variables, k≥2. Explain the geometric signiﬁ- canceof (a) Nosolution. (b) Auniquesolution. (c) Aninﬁnitenumberofsolutions. 1.2SystemsOfEquations,AlgebraicProcedures Outcomes A. Useelementaryoperationstoﬁndthesolutiontoalinearsystemofequations. B. Findtherow-echelonform andreducedrow-echelonform ofamatrix. C. Determine whether a system of linear equations has no solution, a unique solution or an inﬁnitenumberofsolutionsfromitsrow-echelonform. D. SolveasystemofequationsusingGaussianEliminationandGauss-JordanElimination. E. Modelaphysicalsystemwithlinearequationsandthensolve. Wehavetakenanindepthlookatgraphicalrepresentationsofsystemsofequations,aswellashowto ﬁndpossiblesolutionsgraphically. Ourattentionnowturnstoworkingwithsystemsalgebraically.8 SystemsofEquations Deﬁnition1.2:SystemofLinearEquations Asystemoflinearequations isalistofequations, a x +a x +···+a x =b 11 1 12 2 1n n 1 a x +a x +···+a x =b n 21 1 22 2 2n 2 . . . a x +a x +···+a x =b m1 1 m2 2 mn n m where a and b are real numbers. The above is a system of m equations in the n variables, ij j x ,x ···,x . Written more simply in terms of summation notation, the above can be written in 1 2 n theform n a x =b, i=1,2,3,···,m ij j i ∑ j=1 The relative size of m and n is not important here. Notice that we have allowed a and b to be any ij j real number. We can also call these numbersscalars . We will use this term throughoutthe text, so keep inmindthatthetermscalarjustmeansthatweareworkingwithreal numbers. Now,supposewehaveasystemwhereb =0foralli. Inotherwordseveryequationequals0. Thisis i aspecialtypeofsystem. Deﬁnition1.3:HomogeneousSystemofEquations A system of equations is called homogeneous if each equation in the system is equal to 0. A homogeneoussystemhastheform a x +a x +···+a x =0 11 1 12 2 1n n a x +a x +···+a x =0 21 1 22 2 2n n . . . a x +a x +···+a x =0 m1 1 m2 2 mn n wherea arescalarsandx arevariables. ij i Recall from the previous section that our goal when working with systems of linear equations was to ﬁndthepointofintersectionoftheequationswhengraphed. Inotherwords,welookedforthesolutionsto thesystem. Wenowwishtoﬁndthesesolutionsalgebraically. Wewanttoﬁndvaluesforx ,···,x which 1 n solvealloftheequations. Ifsuchasetofvaluesexists,wecall (x ,···,x )thesolutionset. 1 n Recall the abovediscussions about the types of solutionspossible. We will see that systems of linear equationswillhaveoneuniquesolution,inﬁnitelymanysolutions,ornosolution. Considerthefollowing deﬁnition. Deﬁnition1.4:ConsistentandInconsistentSystems A system of linear equations is called consistent if there exists at least one solution. It is called inconsistentifthereisnosolution.

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