Find the monic generators of the ideals

Question

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 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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Not too sure about this part...

The eigenvalues of the matrix is only $c=0$. So the eigenspace is spanned by $\epsilon_4$.

Answer 1

I guess, for a given $T:V\to V$ and $W\le V$, the set $S(v,W)$ is defined as $$S(v,W)\ :=\ \{f\in F[x]:f(T)v\in W\}$$ Now this is an ideal of the polynomial ring $F[x]$, if $\ W$ is a $T$-invariant subspace, because then $f(T)v\in W\implies T(f(T)v)\in W$

which means $f\in S(v,W)\implies x\cdot f\in S(v,W)$, so in turn, $x^n\cdot f\in S(v,W)$ and by their linear combinations we arrive to $g\cdot f\in S(v,W)$ for any $g\in F[x]$.

In our case, $W$ is the eigenspace of $T$ for eigenvalue $c$, hence it is $T$-invariant.

Since $F[x]$ is a principal ideal domain, as an ideal, $S(v,W)$ is generated by a single polynomial. The word 'monic' means only that the lead coefficient is $1$, which is not a big deal, as $F$ is a field.

Finding $S(e_4,W)$ is easy: already $f(x)=1$ satisfies $Ie_4\in W$.

We'll get $S(e_3,W)=(x-c)\ $ because $(T-cI)e_3\in W$ but no constant polynomial satisfies this.

For part a), you also have to prove that no vector $v\notin W$ makes $(T-cI)v=0$.