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.
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
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