# 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.

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andrewkirk
I don't know what they mean by $S(\epsilon_j;W)$ but usually problems about matrices and monic generators of ideals are concerned with the minimal polynomial of a matrix, and the ideals in question are ideals of the polynomial ring $F[x]$, which will be a principal ideal domain, so that any ideal of it can be generated by a single monic polynomial. The minimal polynomial of a matrix A is the unique monic polynomial that can generate the ideal consisting of all polynomials $p[x]$ in $F[x]$ such that $f[A]=0$. Here 'monic' means that the highest-order coefficient of the polynomial is 1.