In one of my questions, we have $L: P_3(\mathbb{R}) \to \mathbb{R}^3$, and $L$ is defined as $L(p) = \left( \begin{array}{c} p(-2) \\ p(1) \\ p(3) \end{array} \right)$. What does $L(p)$ mean? Since the basis for $P_3(\mathbb{R})$ is $\{1, x, x^2, x^3\}$, then does it mean that for $L(1), L(x), L(x^2),$ and $L(x^3)$, everything will equal to $-2, 1, 3$ regardless of the variable given?
edit: the question is to find a matrix representing the linear transformation $L$, with bases $\{1, x, x^2, x^3\}$ and the standard basis for $\mathbb{R}^3$, $\{e_1, e_2, e_3\}$. I know how to do these kinds of questions, I'm just not sure what $L(p)$ is in this case.
Actually I'm unsure about this. Is the following solution correct?
$ L(1) = (-2, 1, 3) \\ L(x) = (-2, 1, 3) \\ L(x^2) = (4, 1, 9) \\ L(x^3) = (-8, 1, 27) $
$A = \left( \begin{array}{cccc} -2 & -2 & 4 & -8 \\ 1 & 1 & 1 & 1 \\ 3 & 3 & 9 & 27 \end{array} \right) $