MA554_JAN97 - Qualifying Examination January 1997 Math 554...

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Qualifying Examination January, 1997 Math 554 In answering any part of a question you may assume the preceding parts. Notation: V is a finite dimensional vector space over a field K ; α : V -→ V is a linear operator. 1. In some basis of V , α is given by the matrix A = 1 1 - 2 0 2 1 0 2 1 0 1 1 0 - 1 2 1 . Find: (1) the rational normal form of α . [8] (2) the Jordan normal form of α . [7] 2. Let P be the space of polynomials of degree < n over K , and let δ : P -→ P be the operator, given by differentiation: δ ( ∑ n - 1 i =0 a i x i ) = n - 1 i =1 ia i x i - 1 . Find the Jordan normal form of the δ 2 , when (1) K is the field C of complex numbers. [5] (2) K is the field F 3 with 3 elements. [5] 3. A α -invariant subspace W V is called irreducible, if the only proper α - invariant subspaces of W are 0 and W itself. (1) Prove that if the characteristic polynomial of α has an irreducible factor of degree d , then α has an irreducible invariant subspace of dimension d . [10] (2) Prove the converse of (1). [10] 4. Let v 1 , v 2 and w 1 , w 2 be two pairs of vectors in a real inner product space V . (1) Prove that if || v 1 || = || w 1 || , || v 2 || = || w 2 || , and (
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