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Unformatted text preview: of T which is larger than 1. (Hint: Spectral Theorem) 5. (20 points) Let V be a complex vector space. If you get stuck on part (a) below, assume it is true and use it in part (b). (a) Prove that if N is a nilpotent operator on V , then N + I has a square root. (b) Prove that any invertible operator T on V has a square root. (Hint: Use the generalized eigenspaces of T ) 6. (15 points) Suppose that an operator T on a complex vector space has characteristic polynomial z 3 ( z2) 5 ( z + 1) 2 and minimal polynomial of the form z 2 ( z2) k ( z + 1) ‘ where k > 2 and ‘ ≥ 1 . Suppose further that dim range( T2 I ) = 7 and that the eigenspace corresponding to1 is 1dimensional. Find, with justiﬁcation, the Jordan blocks which make up the Jordan form of T . You do not have to write out the full Jordan form itself. 7. (0 points) Draw a picture portraying your love of linear algebra....
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 Spring '08
 GUREVITCH
 Math, Linear Algebra, innerproduct space

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