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Unformatted text preview: ALGEBRA PRELIMINARY EXAMINATION - JANUARY 2002 Let A be a matrix and assume A2 has'characteristic polynomial x3(x-l)2 and minimal polynomial x2 (x —' 1). What are the possible Jordan canonical forms of A ? Let T: V —> W be a linear transformation between two vector spaces Vand W. Show that T is injective if and only if Ker(T) = {v e V I T(v) = 0} only contains the vector 0. Let T: V —) W be a linear transformation between two finite dimensional vector spaces Vand W. Show that T is an isomorphism if and only if the dual map T‘ : W‘ —) V* is an isomorphism. Let T: V—) V be a linear operator on a vector spaces V and assume v1,v2,...,vk are eigenvectors of T corresponding to. the distinct eigenvalues a1,oc2,...,ak. Show that v1,v2,. . .,vk are linearly-independent. Suppose A is an an matrix over the real numbers R. Show that A is diagonalizable over R if and only if we can find a basis for R" consisting of eigenvectors for A. (a) Assume T is a normal linear operator on a finite dimensional complex inner product vector space. Show that eigenvectors corresponding to distinct eigenvalues are orthogonal. (b) Show by example that this need not be true if T is not normal. ...
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