# mathhjhj - 2(15 points Give examples with brief...

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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.
4. (20 points) Suppose that T is a self-adjoint operator on a inner-product space V such that there exists v 2 V with k v k = 1 such that h Tv, v i > 1. Prove that there exists an eigenvalue of T which is larger than 1. (Hint: Spectral Theorem)
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