MA554_JAN03

# MA554_JAN03 - Math 554(12 Qualifying Exam Heinzer January 7...

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Math 554 Qualifying Exam Heinzer January 7, 2003 (12) 1. Let F be a ﬁeld, let n be a positive integer, and let W = F n × n denote the vector space of n × n matrices with entries in F . (i) Let W 0 denote the subspace of W spanned by the matrices C of the form C = AB - BA .Wha ti sd im W 0 ? (ii) For A F n × n , deﬁne the adjoint matrix adj A F n × n . (iii) If A R 3 × 3 and det A =2 ,whatisdetadj A ? (6) 2. Let T : C 5 C 5 be a linear operator and let g ( x ) be a polynomial in C [ x ]. If c is a characteristic value for g ( T ), must there exist a characteristic value a for T such that g ( a )= c ? Explain why or why not. 1

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2 (20) 3. Let A C 4 × 4 be a diagonal matrix with main diagonal entries 1 , 2 , 3 , 4. Deﬁne T A : C 4 × 4 C 4 × 4 by T A ( B )= AB - BA . (i) What is the dimension of the null space of T A ? (ii) What is the dimension of the range of T A ? (iii) What are the characteristic values of T A ? (iv) What is the minimal polynomial of T A ? (v) Is T A diagonalizable? Explain.
3 (16) 4. Let F be a ﬁeld, let m and n be positive integers and let A F m × n be an m × n matrix. (i) Deﬁne “row space of A ”. (ii) Deﬁne “column space of A ”. (iii) Prove that the dimension of the row space of A is equal to the dimension of the column space of A .

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4 (16) 5. Let D be a principal ideal domain and let V and W denote free D -modules of rank 4 and 5, respectively. Assume that
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MA554_JAN03 - Math 554(12 Qualifying Exam Heinzer January 7...

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