Linearity coefficient mapping to polynomials. In this Vandermonde matrix section right at the beginning of my book, the authors comment on the linearity properties of polynomials and conclude that this must mean a mapping from polynomial coefficients to sampled polynomial values is linear. Their explanation in photo:

That seems like a large jump to me. Can someone please elaborate on how the coefficients, say, $c_0,...,c_k, \, 1 < k < n-1$ can be linearly mapped as mentioned, to a corresponding polynomial of degree $k$? Importantly, because of the text leading up to that conclusion.
I think I'm just overthinking this. Thank you.
 A: Think of it this way:
Let $p(x)=c_2x^2+c_1x+c_0$. Then we want to create a map that sends $$\mathbf{c}=\begin{bmatrix}c_0\\c_1\\c_2\end{bmatrix} \to \begin{bmatrix}p(x_1)\\p(x_2)\\p(x_3)\end{bmatrix}=\begin{bmatrix}c_0+c_1x_1+c_2x_1^2\\c_0+c_1x_2+c_2x_2^2\\c_0+c_1x_3+c_2x_3^2\end{bmatrix}=\mathbf{p}.$$
But
$$\begin{bmatrix}c_0+c_1x_1+c_2x_1^2\\c_0+c_1x_2+c_2x_2^2\\c_0+c_1x_3+c_2x_3^2\end{bmatrix}=\begin{bmatrix}1&x_1&x_1^2\\1&x_2&x_2^2\\1&x_3&x_3^2\end{bmatrix}\begin{bmatrix}c_0\\c_1\\c_2\end{bmatrix}.$$
So this map is simply a multiplication by (the Vandermonde) matrix, hence linear.
Added note
Based on your comment I think this is what you might be looking for.
Let $T$ be the map from the coefficient space to the the sampled points space. Continuing with the example above, let $q(x)=d_2x^2+d_1x+d_0$, then the map $T$ will send
$$\mathbf{d}=\begin{bmatrix}d_0\\d_1\\d_2\end{bmatrix} \to \begin{bmatrix}q(x_1)\\q(x_2)\\q(x_3)\end{bmatrix}=\mathbf{q}.$$
Now consider
$$T(\mathbf{c}+\mathbf{d})=\begin{bmatrix}(p+q)(x_1)\\(p+q)(x_2)\\(p+q)(x_3)\end{bmatrix}=\begin{bmatrix}p(x_1)+q(x_1)\\p(x_2)+q(x_2)\\p(x_3)+q(x_3)\end{bmatrix}=\begin{bmatrix}p(x_1)\\p(x_2)\\p(x_3)\end{bmatrix}+\begin{bmatrix}q(x_1)\\q(x_2)\\q(x_3)\end{bmatrix}=T(\mathbf{c})+T(\mathbf{d}).$$
Similarly you can check that $T(\lambda \mathbf{c})=\lambda T(\mathbf{c})$. Hence the linearity is obtained based on the linearity of the polynomial space.
