Linear algebra-characteristic and minimal polynomials

Question

Problems:

Linear Algebra : Modules, Linear Operators, Characteristic and Minimal Polynomials, Generators, Abelian Groups and Annihilators

1. Let F be a field and Let  R = F[x]be the polynomial ring over F. What does it mean for an Abelian group M to be a module over F[x]?

2.  Let M be an R-module. The annihilator of M in R is defined by Ann_R(M) = {r ∈ R : rm = 0, ∀ m∈M}.Show that Ann_R(M) is an ideal of R.

3. Let F be a field and n ∈ N. Then Fⁿ is an n-dimensional vector space over F. Define a function T : Fⁿ → Fⁿ by T (y₁,...y_n) = (0,y₁,....,y_n-1).

A. Show that T is a linear operator.

B. Find the characteristic and minimal polynomials for T, with explanation. (For the characteristic polynomial, recall that you will need to choose a basis for Fⁿ , find the matrix of T relative to that basis, and find the characteristic polynomial of the matrix.)

C. By Example 47, we can use T to make Fⁿ into a module over the polynomial ring F[x]. Show that is cyclic F[x] by giving a generator for M, with explanation. Find the Ann_F[x](M)