I have a linear operator $T: \mathbb{R}^4\to\mathbb{R}^4$, $I$ is identity operator.
I am considering subspaces like these.
$V_0=\{0\}\hspace{1cm}V_{i+1}=TV_i+\text{im}(B)$ where $B$ is any other operator on the same space.
my question is if the degree of minimal polynomial of $T$ is given which ofcourse will be $1/2/3/4$ what can we say about $i$ for which $V_i=V_{i+k}\forall k=1,2,\dots$ For example If the degree is $3$ can I say $V_3=V_4=V_5\dots$?
I can see $V_i\subseteq V_{i+1}$, i.e they are nested subspaces inside $\mathbb{R}^4$ right?And due to finite dimension, so the iteration will stop.