Let $V_4 = \{p(x) \in \mathbb{R}[X], \text{ such that } \deg{p(x)} \leq 4\}.$
Consider the linear map $f: V_4 \longrightarrow \mathbb{R}^2$ given by $ p(x) \longmapsto \begin{bmatrix} p(1)\\\ p(2)\end{bmatrix}.$
Show that $\{(x-1)(x-2), x(x-1)(x-2), x^2(x-1)(x-2)\}$ is a basis of the nullspace $U$ of $f$.
Show that $B = \{1, x, (x-1)(x-2), x(x-1)(x-2), x^2(x-1)(x-2)\}$ is a basis of $V_4$
Compute the matrix of $f$ with respect to the basis $B$ of $V_4$ and $C =$ standard basis $\{(1,0), (0,1)\}$ of $\mathbb{R}^2$. I know the answer is: $\begin{bmatrix} 1 & 1 & 0 & 0 & 0 \\\ 1 & 2 & 0 & 0 & 0 \end{bmatrix}$ but I don't know how to reach this answer. It has something to do with computing $f(1), f(x)$ etc, which equal $(1,1)$ and $(1,2)$, respectively, but I don't understand how you work this out.