Determine transformation matrix "A linear transformation between polynomial spaces: $f: P_2(\mathbb{R}) \to P_2(\mathbb{R})$ is given by: $$f(p(x))=3 \cdot p(1)-x^2 \cdot p(0)+(x-1) \cdot p'(1)$$
Provide the transformation matrix with respect to the monomial basis $(1,x,x^2)$
How can one determine this?
 A: A linear transformation can always be represented as a matrix, once you have chosen a basis for the domain and codomain. Since $f$ is a map from a three-dimensional vector space to itself, it has a representation as a $3\times 3$ matrix:
$$\left[\begin{array}{c} f(p)_1 \\ f(p)_2 \\ f(p)_3\end{array}\right] = \left[\begin{array}{ccc}? & ? & ?\\? & ? & ?\\?&?&?\end{array}\right]\left[\begin{array}{c}p_1\\p_2\\p_3\end{array}\right]$$
where the argument to $f$ is $p(x) = p_1 + p_2 x + p_3 x^2$ and the result is $f(p(x)) = f(p)_1 + f(p)_2x + f(p)_3x^2$.
Now can you "probe" the '?'s by plugging in different polynomials into $f$? For example, if you plug in $p(x) = 1$, which in the monomial basis is the vector $\left[\begin{array}{c}1 \\ 0 \\ 0\end{array}\right]$, what is $f(p)$, in the monomial basis? What does this tell you about the entries of the matrix? How do you find the rest of the '?'s?
A: Find $f(1), f(x)$ and $f(x^2) $. Take the coefficients of  $1,x$ and $x^2$  in $f(1)$ and turn it into a column.
For example $f(1) =3-x^2$ so the first column of the transformation matrix 
$\left(
\begin{array}{c}
3\\
0\\
-1\\
\end{array}
\right)$
Do the same for $f(x)$ and $f(x^2)$ to get the second and third column of transformation matrix.
