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# final - of T which is larger than 1(Hint Spectral Theorem...

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MATH 110 - FINAL LECTURE 1, SUMMER 2009 August 13, 2009 Name : 1. (10 points) Give two equivalent deﬁnitions (or characterizations) of each of the following. (a) A normal operator on an inner-product space V . (b) A generalized eigenvector of an operator T . (c) A positive operator on an inner-product space V . (d) An isometry on an inner-product space V .

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2. (15 points) Give examples, with brief justiﬁcation, of each of the following. (a) An operator on R 2 which is not self-adjoint with respect to the standard inner product. (b) An isometry on R 4 with no (real) eigenvalues. (c) An operator on C 4 whose characteristic polynomial equals the square of its minimal polynomial.
3. (20 points) Suppose that P is an operator on an inner-product space V such that P 2 = P . Prove that P is an orthogonal projection if and only if it is self-adjoint.

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4. (20 points) Suppose that T is a self-adjoint operator on a inner-product space V such that there exists v V with k v k = 1 such that h Tv,v i > 1. Prove that there exists an eigenvalue

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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 ( z-2) 5 ( z + 1) 2 and minimal polynomial of the form z 2 ( z-2) k ( z + 1) ‘ where k > 2 and ‘ ≥ 1 . Suppose further that dim range( T-2 I ) = 7 and that the eigenspace corresponding to-1 is 1-dimensional. 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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final - of T which is larger than 1(Hint Spectral Theorem...

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