南京航空航天大学Matrix-Theory双语矩阵论期末考试2015.doc

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1、第5页（共5页）PartI(必做题，共5题，70分)第1题（15分）得分Letdenotethesetofallrealpolynomialsofdegreelessthan3withdomain（定义域）.Theadditionandscalarmultiplicationaredefinedintheusualway.Defineaninnerproductonby.(1)ConstructanorthonormalbasisforfromthebasisbyusingtheGram-Schmid

2、torthogonalizationprocess.(2)Let.Findtheprojectionofontothesubspacespannedby{}.Solution:(1),,,,,-------------------------------------------------------------------------------------------(2)---------------------------------------------------------------

3、-------------------------------------------------第5页（共5页）第2题（15分）得分Letbethelineartransformationon(thevectorspaceofrealpolynomialsofdegreelessthan3)definedby.(1)Findthematrixrepresentingwithrespecttotheorderedbasis[]for.(2)Findabasisforsuchthatwithrespec

4、ttothisbasis,thematrixBrepresentingisdiagonal.(3)Findthekernel（核）andrange（值域）ofthistransformation.Solution:(1)-----------------------------------------------------------------------------------------------------------------(2)(ThecolumnvectorsofTarethee

5、igenvectorsofA)Thecorrespondingeigenvectorsinare(TdiagonalizesA).Withrespecttothisnewbasis,therepresentingmatrixofisdiagonal.-------------------------------------------------------------------------------------------------------------------(3)Thekerneli

6、sthesubspaceconsistingofallconstantpolynomials.Therangeisthesubspacespannedbythevectors-----------------------------------------------------------------------------------------------------------------------第5页（共5页）第3题（20分）得分Let.(1)Findalldeterminantdivi

7、sorsandelementarydivisorsof.(2)FindaJordancanonicalformof.(3)Compute.(Givethedetailsofyourcomputations.)Solution:(1),(特征多项式.Eigenvaluesare1,2,2.)Determinantdivisoroforder,,Elementarydivisorsare.-----------------------------------------------------------

8、-----------------------------------------------------------(2)TheJordancanonicalformis--------------------------------------------------------------------------------------------------------------------------(3)Foreigenvalue1,，An