Qualifying ExaminationAugust, 1996Math 554In answering any part of a question you may assume the preceding parts.Notation:Kis a ﬁeld;Mn(K) is the set ofn×nmatrices with elements fromK;Vis ann-dimensional vector space overK;αis a linear operator onV.1.Prove that ifA,B∈Mn(K) and one ofis invertible, then det(aA+B)=0for at mostndistinct values ofa∈K.[8points]2.LetATdenote the transpose ofA∈Mn(K). Prove that there exists an invertibleP∈Mn(K), such thatPAP-1=AT8points]3.Letπ1andπ2be linear operators on a vector spaceV, such thatπ1π2=π2π1π21=π1π22=π2.Prove thatVis the direct sum of the following four subspaces:[8 points]Imπ1∩Imπ2Imπ1∩Kerπ2Kerπ1∩Imπ2Kerπ1∩Kerπ2.4.Prove that ifαhas the same matrix in all bases ofV, then there exists ana∈Ksuch thatα=aidV8points]5.Prove that if rank(α) = 1, then the minimal polynomial ofαhas the formx(x-a)forsomea∈
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