1. ## Finding eigenvalues

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 $T^2v$.

Exercise: Suppose the linear map $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ is given by $T(a,b)=(a+4b,3a+5b)$.
Let $v=(2,1)$. Find coefficients $a_0,a_1,a_2$ so that $a_{0}v+a_{1}Tv+a_{2}T^{2}v=0$.

2. Originally Posted by zebra2147
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 $T^2v$.

Exercise: Suppose the linear map $T:\mathbb{R}^2 \rightarrow \mathbb{R}^2$ is given by $T(a,b)=(a+4b,3a+5b)$.
Let $v=(2,1)$. Find coefficients $a_0,a_1,a_2$ so that $a_{0}v+a_{1}Tv+a_{2}T^{2}v=0$.
$T^2v$ means $T(T(v))$, so that

$T^2(a,b)=T(T(a,b))=T(a+4b,3a+5b)$
$=(a+4b+4(3a+5b),3(a+4b)+5(3a+5b))$
$=(13a+24b,18a+37b)$.