MA554_AUG96 - Qualifying Examination August, 1996 Math 554...

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Qualifying Examination August, 1996 Math 554 In answering any part of a question you may assume the preceding parts. Notation: K is a field; M n ( K ) is the set of n × n matrices with elements from K ; V is an n -dimensional vector space over K ; α is a linear operator on V . 1. Prove that if A,B M n ( K ) and one of is invertible, then det( aA + B )=0 for at most n distinct values of a K .[ 8 p o i n t s ] 2. Let A T denote the transpose of A M n ( K ). Prove that there exists an invertible P M n ( K ), such that PAP - 1 = A T 8 p o i n t s ] 3. Let π 1 and π 2 be linear operators on a vector space V , such that π 1 π 2 = π 2 π 1 π 2 1 = π 1 π 2 2 = π 2 . Prove that V is the direct sum of the following four subspaces: [8 points] Im π 1 Im π 2 Im π 1 Ker π 2 Ker π 1 Im π 2 Ker π 1 Ker π 2 . 4. Prove that if α has the same matrix in all bases of V , then there exists an a K such that α = a id V 8 p o i n t s ] 5. Prove that if rank( α ) = 1, then the minimal polynomial of α has the form x ( x - a )forsome a
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