# Find monic generators of the ideals

## Homework Statement

Let ##T## be the linear operator on ##F^4## represented in the standard basis by $$\begin{bmatrix}c & 0 & 0 & 0 \\ 1 & c & 0 & 0 \\ 0 & 1 & c &0 \\ 0 & 0 & 1 & c \end{bmatrix}.$$ Let ##W## be the null space of ##T-cI##.

a) Prove that ##W## is the subspace spanned by ##\epsilon_4##.

b) Find the monic generators of the ideals ##S(\epsilon_4;W),\,S(\epsilon_3;W),\,S(\epsilon_2;W)##, and ##S(\epsilon_1;W)##.

## The Attempt at a Solution

The first part is easy. It's trivial to see that ##T-cI## sends vectors of the form ##(0,0,0,d)## to ##0##, such that the null space is spanned by ##\epsilon_4=(0,0,0,1)##. However, I have no idea how to start the second part. I'm having some trouble understanding what is meant by ##S(\epsilon_i;W)##. Any help would be appreciated.

andrewkirk