MA554_JAN97

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 = 11 - 20 21 02 10 0 - 121 . 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 ,andlet 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 ( v 1 2 )= ( w 1 2 ), then there is
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