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# Algebra1992Aug - PhD Preliminary Examination Algebra Fall...

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Unformatted text preview: PhD. Preliminary Examination Algebra Fall 1992 1. Let T: V—9 W be alinear transformation of finite dimensional vector spaces. Assume that rank 1‘: k. Prove that there exist ordered bases B for V, and C for W. such that the matrix representation of T with respect to B and C has the following property : its (i,i) entry equals one for i= 1,2...k , and all its other entries are zero. 2. Suppose V = W1 \$ W2 and that f1 and f2 are inner products on W1 and W2 respectively. Show that there is a unique inner product f on V such that (a) w2 = WJ' ; (b) f(0t,B ) = fk (aﬁ ) when 0t,]3 are in Wk , k: 1,2. 3. Let V be an n—dimensional vector space and let T be a linear operator on V. Suppose that there exists a positive integer k such that Tk = O. Prove that Tn = 0. What is the characteristic polynomial for T ? -3 l -l 4. Suppose B: -—7 5 -1 . Find:(a) the characteristic polynomial and the ——6 6 2 eigenvalues of B; and (b) a maximal set S of linearly independent eigenvectors of B. (c) Is B diagonalizable ? 5. If A is a square matrix with characteristic polynomial f(x)=(x—2)3 (x+3)4 and minimal polynomial g(x) = (x-2)(x-t-3)2 , give all possible Jordan normal forms for A. 6. Let T:V-—>W be a linear transformation with dim V=n, dim W: m, and rank T: k. Let Tt : W —-)V‘ be the dual linear transformation. What are the rank and the nullity of 'I‘ ? ...
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