Assume all rings are commutative with 1 not equal to 0 unless specifically stated otherwise. Assume that k[t] is a polynomial ring over a field k.

- Let f(t) be a polynomial in k[t] of degree at least 1. Show that the vector space dimension of V = k[t]/(f(t)) is deg(f).
- Let f(t) be a nonzero monic polynomial in k[t].
- (a) Let T: k[t]/(f(t)) -> k[t]/(f(t)) be the linear operator given by multiplication by t. Show that the characteristic polynomial p(t) of T is f(t). (See p. 122 for the definition.) [Hint: find the matrix for T in terms of the basis given by the powers of t.]
- (b) Now let f
_{1}(t), ..., f_{r}(t) be nonzero monic polynomials in k[t]. Let V = k[t]/(f_{1}(t)) x ... x k[t]/(f_{r}(t)). Let T: V -> V be the linear operator given by multiplication by t. Show that the characteristic polynomial p(t) of T is the product f_{1}(t)...f_{r}(t). (You may assume the fact that the determinant of a block diagonal matrix is the product of the determinants of the blocks.)

- Let p(t) be the characteristic polynomial p(t) of
an n x n matrix A with entries in a field k. Let V be
k
^{n}, regarded as a k[t]-module via the action of A.- (a) If f
_{1}(t), ..., f_{r}(t) are the invariant factors for A (i.e., monic, positive degree, each one divides the next), with f_{r}(t) being the factor of largest degree, show that f_{r}(A) = 0. - (b) If f(t) is any polynomial such that
f(A) = 0, show that f
_{r}(t) divides f(t). [Note: this is why f_{r}(t) is called the minimal polynomial of A; f_{r}(t) is the nonzero monic polynomial g(t) of minimum possible degree such that g(A) = 0.] - (c) Conclude from (a) and the previous problem that f
_{r}(t) divides p(t) and that p(A) = 0. [This is known as the Cayley-Hamilton theorem.]

- (a) If f
- Let A be
an n x n matrix with entries in a field k. Let V be
k
^{n}, regarded as a k[t]-module via the action of A. Show that tI - A is a presentation matrix for V. [Hint: Note that we can regard A as also giving a k[t]-module map k[t]^{n}-> k[t]^{n}. Let e_{1}, ..., e_{n}be the standard basis for k[t]^{n}, and let e'_{1}, ..., e'_{n}be the standard basis for V. Let h: k[t]^{n}-> V be the map h(g_{1}(t)e_{1}+ ... + g_{n}(t)e_{n}) = g_{1}(t)e'_{1}+ ... g_{n}(t)e'_{n}. If g_{1}(t)e_{1}+ ... + g_{n}(t)e_{n}is in ker(h), it will be helpful to show that g_{1}(A)e_{1}+ ... + g_{n}(A)e_{n}= 0. It will also be helpful to note that t^{i}e_{j}- A^{i}e_{j}= t^{i-1}(tI - A)e_{j}+ t^{i-2}(tI - A)Ae_{j}+ ... + (tI-A)A^{i-1}e_{j}.] - If A is a 3 x 3 matrix with entries in k and
with characteristic polynomial p(t) =
(t+1)
^{2}(t+2) and minimal polynomial (t+1)(t+2), find the rational canonical form of A. Justify your answer.