Let $\displaystyle T:V\to V$ be linear transformation over field, and let $\displaystyle p$ be a polynomial over that field.
Prove that $\displaystyle p(T)$ nilpotent iff $\displaystyle p(0)=0$
Thanks!
Yes, I'm aware this fact.
Check me please...
i) $\displaystyle p(0)=0 \implies p(T) $ nilpotent.
$\displaystyle p(T)=a_1T+a_2T^2+ \dots +T^n$
Above, sum of nilpotent operators $\displaystyle \implies p(T)$ nilpotent.
ii) $\displaystyle p(T) nilpotent \implies p(0)=0$
So, there exist $\displaystyle r$, index of nilpotents so that$\displaystyle {p(T)}^r=0 \implies a_0=0 $