Let $n \in \mathbb{N}$. We consider $n+1$ distinct points in $I$, denoted $x_0 < x_1 < \cdots < x_n$, and a continuous function $f$ from $I$ to $\mathbb{R}$.
Show that the linear map $\varphi : \left|\,\begin{array}{ccl} \mathbb{R}_n[X] & \rightarrow & \mathbb{R}^{n+1} \\ P & \mapsto & \left(P(x_0), P(x_1), \ldots, P(x_n)\right) \end{array}\right.$ is an isomorphism.