Exercise 8.A.5

Proof. Let $a_0,a_1,\dots ,a_{m-1}\in 𝔽$ such that

Applying $T^{m-1}$ to both sides of the equation above yields

which shows that $a_0=0$. Therefore

Applying $T^{m-2}$ yields

which shows that $a_1=0$. Continuing in this fashion, we see that $a_0=a_1=\cdots =a_m=0$. Thus $v,\mathit{Tv},T^{2}v,\dots ,T^{m-1}v$ is linearly independent. □