The following is an exercise that my professor gave us. I am confident that I can solve it but I'm not sure how to interpret $\displaystyle T^2v$.

Exercise: Suppose the linear map $\displaystyle T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ is given by $\displaystyle T(a,b)=(a+4b,3a+5b)$.

Let $\displaystyle v=(2,1)$. Find coefficients $\displaystyle a_0,a_1,a_2$ so that $\displaystyle a_{0}v+a_{1}Tv+a_{2}T^{2}v=0$.