中山大学~线性代数期末总复习.ppt

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1、REVIEWFORTHEFINALEXAMGaoChengYingSunYat-SenUniversitySpring2007LinearAlgebraandItsApplicationREVIEWFORTHEFINALEXAMChapter1LinearEquationsinLinearAlgebraChapter2MatrixAlgebraChapter3DeterminantsChapter4VectorSpacesChapter5EigenvaluesandEigenvectorsChapter6OrthogonalityandLeastSquaresC

2、hapter7SymmetricMatricesandQuadraticFormsCHAPTER1LinearEquationsinLinearAlgebraChapter1LinearEquationinLinearAlgebra§1.1SystemsofLinearEquations§1.2RowReductionandEchelonForms§1.3VectorEquation§1.4TheMatrixEquationAx=b§1.5SolutionSetsofLinearSystems§1.7LinearIndependence§1.8Introduct

3、iontoLinearTransformation§1.9TheMatrixofaLinearTransformation1.1SystemsofLinearEquations1.linearequationa1x1+a2x2+...+anxn=bSystemsofLinearEquations§1.1SystemsofLinearEquationsConfficientmatrixandaugmentedmatrixCoefficientmatrixaugmentedmatrix§1.1SystemsofLinearEquationsAsolutiontoas

4、ystemofequationsAsystemoflinearequationshaseither1.Nosolution,or2.Exactlyonesolution,or3.Infinitelymanysolutions.consistentinconsistent｝§1.1SystemsofLinearEquationsSolvingaLinearSystemElementaryRowOperations1.(Replacement)Replaceonerowbythesumofitselfandamultipleofanotherrow.2.(Inter

5、change)Interchangetworows.3.(Scaling)Multiplyallentriesinarowbyanonzeroconstant.Examples1.SolvingaLinearSystem2.Discussthesolutionofalinearsystemwhichhasunknownvariable§1.1SystemsofLinearEquationsExistenceandUniquenessQuestionsTwofundamentalquestionsaboutalinearsystem1.Isthesystemcon

6、sistent;thatis,doesatleastonesolutionexist?2.Ifasolutionexists,isittheonlyone;thatis,isthesolutionunique?§1.2RowReductionandEchelonFormsThefollowingmatricesareinechelonform:Thefollowingmatricesareinreducedechelonform:pivotposition§1.2RowReductionandEchelonFormsTheorem1Uniquenessofthe

7、ReducedEchelonFormEachmatrixisrowequivalenttooneandonlyonereducedechelonmatrix.§1.2RowReductionandEchelonFormsTheRowReductionAlgorithmStep1Beginwiththeleftmostnonzerocolumn.Step2Selectanonzeroentryinthepivotcolumnasapivot.Step3Userowreplacementoperationstocreatezerosinallpositionsbel

8、owthepivot.S