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1、线性代数难点解析(Analysisofdifficultiesinlinearalgebra)Ananalysisofdifficultproblemsinlinearalgebra.TxtChapterdeterminantI.emphasis1.Understanding:thedefinitionofadeterminant,acofactor,analgebraiccofactor.2,grasp:determinantofthebasicnatureandinference.3,theuseof:theuseofdeterminantp
2、ropertiesandcalculationmethodstocalculatethedeterminant,usingtheClemruleforsolvingequations.Two,difficultiesTheapplicationofdeterminantinthesolutionoflinearequations,theinverseofmatrices,thelinearcorrelationofvectorsandtheeigenvaluesofmatrices.Threeimportantformula1,ifAisamat
3、rixofN,then,kA/=kn/A/2,ifAandBarenordermatrix,is/AB/A/B/=,,,3,ifAisamatrixofN,then,A*//A/n-1=IfAisninvertiblematrix,then/A-1/A/-1/=4,ifAisnordersquare,lambdaI(i=1,2,...N,A)isthecharacteristicvalueofA/PIlambda=IFour,questionsandSolutions1,thepropositionabouttheconceptandproper
4、tyofdeterminant2,thecalculationofthedeterminant(method)1)usedefined2)reducetheorderofadeterminantaccordingtoarow(column)3)thenatureofthedeterminantRows(columns)addedtothesamerow(column)tobeappliedtotheequalityoftheelementsofeachcolumn(row).Doubleorminusthesameline(column)ofea
5、chrow(column),reducethedeterminant,orturnitintotheupper(lower)triangledeterminant.Third,successive(column)additionandsubtraction,andsimplifieddeterminant.Breakthedeterminantintothesumanddifferenceofseveraldeterminants.4)recursivemethodisapplicabletothedeterminantwithstrongreg
6、ularityandzeroelements5)mathematicalinduction,moreusedtoprove3,useClem'slawtosolvelinearequationsIfD=A//=0,thenAx=bhasauniquesolution,i.e.X1=D1/D,x2=,D2/D,...Xn=Dn/DWhereDjistochangethecoefficientsofXJintoconstantsinD.Note:theClemlawappliesonlytoequationswhosenumberofequation
7、sisequaltothenumberofunknowns.4,usingthecoefficientdeterminantsolutionoftheproblemofdiscriminationofAequations1)when,A/=0,Ax=0homogeneousequationswithnonzerosolution;non-homogeneousequationsAx=Bisnottheonlysolution(mayhavenosolution,mayalsohaveinfinitelymanysolutions)2)when,A
8、/=0,theequationAx=0onlyzerosolution;non-homogeneousequationAx=Bisthe