Finding the minimal polynomial of a given linear operator in $P_3(\mathbb{F})$

Question

I am given that $T:P_3(\mathbb{F})\to P_3(\mathbb{F})$ and that $T(f(x))=f^{'''}(x)+f^{''}(x)+2f(x)$, and I am asked to find the minimal polynomial for this. We were given some theorems about the minimal polynomial but our only example involved some linear operator that was never stated, with a given rational canonical basis for $T$. I am just a bit unsure how to get started with finding this minimal polynomial, if anyone could help out a bit it would be greatly appreciated.

Answer 1

Differentiation is nilpotent on polynomials of a fixed order: $$ f \in \mathbb{P}_{3} \mapsto f' \mapsto f'' \mapsto f''' \mapsto 0. $$ So $Tf = f'''+f''+2f= (D^{3}+D^{2}+2I)f=(L+2I)f$ where $L^{2}= D^{4}(D+I)^{2}=0$. The minimal polynomial is $m(\lambda)=(\lambda-2)^{2}$.