- 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 A
^{n}= 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 O
_{2}with det(A) = -1, prove that A^{2}is unipotent. - (e) If A is in O
_{3}with det(A) = -1, show by example that A^{2}need not be unipotent, but prove that -1 is an eigenvalue of A.

- (a) If A is nilpotent, prove that 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 T
^{i}is T-invariant for every i > 0. - (b) Show that Im T
^{i}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 T
^{n}) = Im T^{n}, and prove that s: (ker T^{n})x(Im T^{n}) -> V is an isomorphism.

- (a) Show that ker T
- 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 T
^{i}, and let K be the union for all i of Ker T^{i}. 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.]

- (a) Prove that S is a subgroup of the reals