I have the following problem
Consider the vector space $\mathbb C^3$, the matrix
$$A=\begin{pmatrix} 0 & 1 & 1\\ 0 & 0 & 1\\ 0 & 0 &0 \end{pmatrix}$$
$\mathbb C^3$ is made into a $\mathbb C[T]$-module where $T$ acts like $A$.
Denote by $L(v)$ the submodule generated by $v$. Find all $v$ with dim$(L(v))$.
I consider $L(v)=v\cdot \mathbb C[T]$ and $L(v)=\text{span}\{u_1,u_2\}$. I know if $w\in L(v)$, then $w=c_0v+c_1Av+c_2A^2v$ and $w=\alpha_1u_1+\alpha_2u_2$.
But I do not have an idea to deal with this to find $u_1,u_2$.