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.)

Let A be an nxn matrix with entries in a field F. We say A is nilpotent if A^m = 0 for some m > 0. We say A is unipotent if (A-I) is nilpotent.
- (a) If A is nilpotent, prove that Aⁿ = 0.
- (b) If A is unipotent prove that the characteristic polynomial of A is (t-1)ⁿ.
- (c) If A is unipotent prove that A is similar to an upper triangular matrix with only 1's on the diagonal.
- (d) If A is in O₂ with det(A) = -1, prove that A² is unipotent.
- (e) If A is in O₃ with det(A) = -1, show by example that A² need not be unipotent, but prove that -1 is an eigenvalue of A.
Let T: V -> V be a linear operator on a vector space V over a field F. Do not assume V is finite dimensional.
- (a) Show that ker Tⁱ is T-invariant for every i > 0.
- (b) Show that Im Tⁱ is T-invariant for every i > 0.
- (c) Show by example that the summing map s: (ker T)x(Im T) -> V defined by s((v,w)) = v+w, need not be an isomorphism, even if V is finite dimensional.
- (d) If dim V = n, show that T(Im Tⁿ) = Im Tⁿ, and prove that s: (ker Tⁿ)x(Im Tⁿ) -> V is an isomorphism.
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.
- (a) Is (*) true if V is finite dimensional? Prove or give a counterexample.
- (b) Is (*) true whether or not V is finite dimensional? Prove or give a counterexample.
Do #18(a) of the Miscellaneous Problems, on p. 154 of Artin.
Do #14 for section 5.2, on p. 189 of Artin.
Let c be a real number. Consider the subset S = {m + nc: m and n are integers} of the reals R.
- (a) Prove that S is a subgroup of the reals R under addition.
- (b) If c is the square root of 2, prove that there is a sequence a_i in S with limit 0. Conclude that S is dense in R (i.e., for each real number r and each real t > 0, there is an element s of S such that |r - s| < t).
- (c) In fact, prove that S is dense in R if and only if c is irrational. [Hint: Use an argument of Dirichlet. For any real r, let f(r) be the fractional part of r; i.e., in decimal notation, the part after the decimal point. Given any integer 1 < n, consider the set X = {f(c), f(2c), f(3c), ..., f(nc)}. Show there are integers i and j with 0 < i < j < n+1 such that 0 < |f(jc) - f(ic)| < 1/(n-1); i.e., that there are two elements of X in the interval [0, 1/(n-1)), or in [1/(n-1), 2/(n-1)), ..., etc.]