Math 817: Problem set 6

Instructions: Write up any four of the following problems (but you should try all of them, and in the end you should be sure you know how to do all of them). Your write ups are due Friday, October 8, 2004. Each problem is worth 10 points, 9 points for correctness and 1 point for communication. (Your goal is not only to give correct answers but to communicate your ideas well. Make sure you use good English, so proofread your solutions. Once you finish a solution, you should restructure awkward sentences, and strike out anything that is not needed in your approach to the problem.)
  1. Let n be a positive integer, let F be a field and let W be a subspace of Fn. Let W' be the set of all v in Fn such that vtw is 0 for all w in W.
  2. The set of all symmetric real nxn matrices is a subspace W of the real vector space Mn(R) of all real nxn matrices. Exhibit a basis and determine the dimension of W.
  3. Problem #5 from 3.4:
  4. Problem #11 from 3.4: Let F = Fp.