$T(a_{0}+a_{1}x+a_{2}x^2)=2a_{0}+a_{2}+(a_{0}+a_{1}+a_{2})x+3a_{2}x^2$- Finding $[T]_{E}^{E}$ Let $T(a_{0}+a_{1}x+a_{2}x^2)=2a_{0}+a_{2}+(a_{0}+a_{1}+a_{2})x+3a_{2}x^2$ be a linear transformation. I need to find the eigen-vectors eigenvalues of $T$.
So, I'm trying to find $[T]_{E}^{E}$  when the base is $E=\{1,x,x^2\}$.
I don't understand how I should use this transformation to do that.
Thanks.

The columns of the matrix you seek are the coordinates of the images under $T$ of the elements of the basis. So you need only compute $T(1)$, $T(x)$, and $T(x^2)$.
