Prove that the map $$\begin{array}{ccl}
\Psi : \mathbb{R}_{K-1}[X] & \rightarrow & \mathbb{R}^K \\
P & \mapsto & \left(P\left(x_1\right), \ldots, P\left(x_K\right)\right)
\end{array}$$ is an isomorphism of vector spaces.
Prove that the map
$$\begin{array}{ccl}
\Psi : \mathbb{R}_{K-1}[X] & \rightarrow & \mathbb{R}^K \\
P & \mapsto & \left(P\left(x_1\right), \ldots, P\left(x_K\right)\right)
\end{array}$$
is an isomorphism of vector spaces.