Let $F$ be a finite-dimensional subspace of $C(\mathbb{R})$, with dimension denoted $p$. For any function $f \in C(\mathbb{R})$ and for any real $x$, we denote $e_{x}(f) = f(x)$. a) Show that there exist real numbers $a_{1}, \ldots, a_{p}$ such that $(e_{a_{1}}, \ldots, e_{a_{p}})$ is a basis of the dual space $F^{*}$. b) If $\left(f_{1}, \ldots, f_{p}\right)$ is a family of elements of $F$, show that $\operatorname{Det}\left(f_{i}\left(a_{j}\right)\right)_{1 \leqslant i,j \leqslant p}$ is non-zero if and only if $\left(f_{1}, \ldots, f_{p}\right)$ is a basis of $F$.
Let $F$ be a finite-dimensional subspace of $C(\mathbb{R})$, with dimension denoted $p$. For any function $f \in C(\mathbb{R})$ and for any real $x$, we denote $e_{x}(f) = f(x)$.\\
a) Show that there exist real numbers $a_{1}, \ldots, a_{p}$ such that $(e_{a_{1}}, \ldots, e_{a_{p}})$ is a basis of the dual space $F^{*}$.\\
b) If $\left(f_{1}, \ldots, f_{p}\right)$ is a family of elements of $F$, show that $\operatorname{Det}\left(f_{i}\left(a_{j}\right)\right)_{1 \leqslant i,j \leqslant p}$ is non-zero if and only if $\left(f_{1}, \ldots, f_{p}\right)$ is a basis of $F$.