Complex polynominals of degree 2 or less

Question

Let $P_3(\mathbb{C})$ be the complex vector space of complex polynomials of degree 2 or less. Let $\alpha,\beta\in\mathbb{C}, \alpha\neq\beta$. Consider the function $L:P_3(\mathbb{C}) \mapsto \mathbb{C}^2$ given by

$L(p)=\begin{bmatrix} p(\alpha) \\ p(\beta)\\ \end{bmatrix},$ for $p\in P_3(\mathbb{C})$

1) Explain how L is a linear transformation

---

I know that for a function to be a linear transformation it must satisfy the following:

$1$. $\{0\}\in L$

$2$. $l_1+l_2=l_3\in L$ (closure under vector addition)

$3$. $cl\in L$ (closure under scalar multiplication)

But as I'm asked to explain and not show how L is a linear transformation I'm thinking I have misunderstood something.

Answer 1

$L$ is not a set ! $L$ is a mapping ! You have to show, that for all $p,q \in P_3(\mathbb{C})$ and all $\delta, \gamma \in \mathbb C$ we have

$L(\delta p + \gamma q)= \delta L(p)+ \gamma L(q)$.