How do I generate my linear transformation matrix in this example? Let $T$ be the linear transformation of space of polynomials $P^3$ given by
$$T(a + bx + cx^2) = a + b(x + 1) + c(x + 1)^2$$
Find all eigenvalues and eigenvectors of $T$.

Hints: $a+bx+cx^2$ is equivalent to the vector $[a,b,c]^{T}$. Expanding the RHS
\begin{align}
T=(a+b+c)+(b+2c)x+cx^2
\end{align}
In terms of matrices, $T$ should be equivalent to
\begin{align}
\begin{bmatrix}
a+b+c \\ b+2c \\ c
\end{bmatrix}=
\begin{bmatrix}
1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
a \\ b \\ c
\end{bmatrix}
\end{align}
Now try to find the eigenvectors. 
