eigenvectors and eigenvalues for finding the matrix representiing T with respcet to $\alpha$ Let $T: \mathcal P_2(\Bbb R)\rightarrow \mathcal P_2(\Bbb R)$ be the linear transformation defined by 
$T(p(x))=p(1)(x^2+x)+p(0)$
Let $\alpha={x^2, x, 1}$ be the standard ordered basis for $\mathcal P_2(\Bbb R)$. Find the matrix representing $T$ with respect to $\alpha$, i.e find $[T]_\alpha$
I let $T(x^2)= x^2+x$, then $[T(x^2)]_\alpha$=$$\left[\begin{matrix}1\\1\\0\end{matrix}\right]  $$
similarly, $T(x)= x^2+x$, $T(1)=x^2+x$, the $[T(x)]_\alpha, T[1]_\alpha$ are all the same as the first one, so I have 
$[T]_\alpha$=$$\left[
    \begin{matrix}
    1 & 1 & 1 \\
    1 & 1 & 1 \\
    0 & 0 & 0 \\
    \end{matrix}\right]
$$  Does that seem right?

Almost. Actually, $T(1)=x^2+x+1$. Therefore$$[T]_\alpha=\begin{bmatrix}1&1&1\\1&1&1\\0&0&1\end{bmatrix}.$$
