Math 817: Problem set 9

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, November 5, 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 A be an nxn matrix with entries in a field F. We say A is nilpotent if Am = 0 for some m > 0. We say A is unipotent if (A-I) is nilpotent.
  2. Let T: V -> V be a linear operator on a vector space V over a field F. Do not assume V is finite dimensional.
  3. Let T: V -> V be a linear operator on a vector space V over a field F. Do not assume V is finite dimensional. Let J be the intersection for all i of Im Ti, and let K be the union for all i of Ker Ti. Consider the statement
    (*) the summing map s: KxJ -> V is an isomorphism.
  4. Do #18(a) of the Miscellaneous Problems, on p. 154 of Artin.
  5. Do #14 for section 5.2, on p. 189 of Artin.
  6. Let c be a real number. Consider the subset S = {m + nc: m and n are integers} of the reals R.