REVIEWFORTHEFINALEXAMGaoChengYingSunYat-SenUniversitySpring2007LinearAlgebraandItsApplicationREVIEWFORTHEFINALEXAMChapter1LinearEquationsinLinearAlgebraChapter2MatrixAlgebraChapter3DeterminantsChapter4VectorSpacesChapter5EigenvaluesandEigenvectorsChapter6OrthogonalityandLeastSquaresChapter7SymmetricMatricesandQuadraticFormsCHAPTER1LinearEquationsinLinearAlgebraChapter1LinearEquationinLinearAlgebra§1.1SystemsofLinearEquations§1.2RowReductionandEchelonForms§1.3VectorEquation§1.4TheMatrixEquationAx=b§1.5SolutionSetsofLinearSystems§1.7LinearIndependence§1.8IntroductiontoLinearTransformation§1.9TheMatrixofaLinearTransformation1.1SystemsofLinearEquations1.linearequationa1x1+a2x2+...+anxn=bSystemsofLinearEquationsmnmnmmnnnnbxaxaxabxaxaxabxaxaxa22112222212111212111§1.1SystemsofLinearEquationsConfficientmatrixandaugmentedmatrixCoefficientmatrixaugmentedmatrixmnmmnnaaaaaaaaa212222111211mmnmmnnbaaabaaabaaa21222221111211§1.1SystemsofLinearEquationsAsolutiontoasystemofequationsAsystemoflinearequationshaseither1.Nosolution,or2.Exactlyonesolution,or3.Infinitelymanysolutions.consistentinconsistent}§1.1SystemsofLinearEquationsSolvingaLinearSystemElementaryRowOperations1.(Replacement)Replaceonerowbythesumofitselfandamultipleofanotherrow.2.(Interchange)Interchangetworows.3.(Scaling)Multiplyallentriesinarowbyanonzeroconstant.Examples1.SolvingaLinearSystem2.Discussthesolutionofalinearsystemwhichhasunknownvariable§1.1SystemsofLinearEquationsExistenceandUniquenessQuestionsTwofundamentalquestionsaboutalinearsystem1.Isthesystemconsistent;thatis,doesatleastonesolutionexist?2.Ifasolutionexists,isittheonlyone;thatis,isthesolutionunique?§1.2RowReductionandEchelonFormsThefollowingmatricesareinechelonform:Thefollowingmatricesareinreducedechelonform:pivotposition§1.2RowReductionandEchelonFormsTheorem1UniquenessoftheReducedEchelonFormEachmatrixisrowequivalenttooneandonlyonereducedechelonmatrix.§1.2RowReductionandEchelonFormsTheRowReductionAlgorithmStep1Beginwiththeleftmostnonzerocolumn.Step2Selectanonzeroentryinthepivotcolumnasapivot.Step3Userowreplacementoperationstocreatezerosinallpositionsbelowthepivot.Step4Applysteps1-3tothesubmatrixthatremains.Repeattheprocessuntiltherearenomorenonzerorowstomodify.Step5Beginningwiththerightmostpivotandworkingupwardandtotheleft,createzerosaboveeachpivot.§1.2RowReductionandEchelonFormsSolutionofLinearSystems(UsingRowReduction)eg.FindthegeneralsolutionofthefollowinglinearsystemSolution:§1.2RowReductionandEchelonFormsTheassociatedsystemnowisThegeneralsolutionis:§1.2RowReductionandEchelonFormsTheorem2ExistenceandUniquenessTheoremAlinearsystemisconsistentifandonlyiftherightmostcolumnoftheaugmentedmatrixisnotapivotcolumn–thatis,ifandonlyifanechelonformoftheaugmentedmatrixhasnorowoftheform§1.3VectorEquationsAlgebraicPropertiesofForallu,v,winandallscalarscandd:where–udenotes(-1)u§1.3VectorEquationsSubsetof-Span{v1,…,vp}iscollectionofallvectorsthatcanbewrittenintheformwithc1,…,cpscalars.§1.4TheMatrixEquationAx=b1.DefinitionIfAisanm×nmatrix,withcolumna1,…,an,andifxisinRn,thentheproductofAandx,denotedbyAx,isthelinearcombinationofthecolumnsofAusingthecorrespondingentriesinxasweights;thatis:§1.4TheMatrixEquationAx=bTheorem3IfAisanm×nmatrix,withcolumna1,…,an,andifbisinRm,thematrixequationAx=bhasthesamesolutionsetasthevectorequationwhich,inturn,hasthesamesolutionsetasthesystemoflinearequationwhoseaugmentedmatrixis§1.4TheMatrixEquationAx=b2.ExistenceofSolutionsTheequationAx=bhasasolutionifandonlyifbisalinearcombinationofcolumnsofA.Example.IstheequationAx=bconsistentforallpossibleb1,b2,b3?§1.4TheMatrixEquationAx=bSolutionRowreducetheaugmentedmatrixforAx=b:∵∴TheequationAx=bisnotconsistentforeveryb.=≠0(forsomechoicesofb)§1.4TheMatrixEquationAx=bTheorem4LetAbeanm×nmatrix.Thenthefollowingstatementsarelogicallyequivalent.Thatis,foraparticularA,eithertheyarealltruestatementsortheyareallfalse.a.ForeachbinRm,theequationAx=bhasasolution.b.EachbinRmisalinearcombinationofthecolumnsofA.c.ThecolumnsofAspanRm.d.Ahasapivotpositionineveryrow.§1.4TheMatrixEquationAx=b3.ComputationofAxExample.ComputeAx,whereSolution.§1.4TheMatrixEquationAx=b4.PropertiesoftheMatrix-VectorProductAxTheorem5IfAisanm×nmatrix,uandvarevectorsinRn,andcisascalar,then:§1.5SolutionSetofLinearSystems1.SolutionofHomogeneousLinearSystems2.SolutionofNonhomogeneousSystems§1.5SolutionSetofLinearSystems1.HomogeneousLinearSystemsAx=0-trivialsolution(平凡解)-nontrivialsolution(非平凡解)ThehomogeneousequationAx=0hasanontrivialsolutionifandonlyiftheequationhasatleastonefreevariable.§1.5SolutionSetofLinearSystemsExampleSolvetheHomogeneousLinearSystemsSolution(1)RowreductionExample(2)Rowreductiontoreducedechelonform(3)Thegeneralsolution§1.5SolutionSetofLinearSystems2.SolutionofNonhomogeneousSystemseg.DescribeallsolutionsofAx=b,whereSolution§1.5SolutionSetofLinearSystemsThegeneralsolutionofAx=bhastheformThesolutionsetofAx=binparametricvectorform§1.5SolutionSetofLinearSystemsTheorem6Suppose