Exercise 8.C.15

Proof. (a) Just repeat the proof of 8.40 replacing $I$ with $v$, $T^j$ with $T^{j}v$ and $n^2$ with $n$.

(b) This is essentially the same as the proof of 8.46.

Let $q$ denote the minimal polynomial of $T$. Then we also have $q(T)v=0$. Furthermore, $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->q\ge \mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->p$. By the Division Algorithm for Polynomials (4.8), there exist $s,r\in \mathbb{P}(𝔽)$ such that

and $\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->r<\mathrm{deg}<!--\; FUNCTION\; APPLICATION\; -->p$. This implies that

By the same reasoning used in the proof, it follows that $r=0$. □