Let $m \in \mathbb{N}$. We consider the matrix
$$M = \left(\begin{array}{cccccc} \binom{0}{0} & 0 & \cdots & \cdots & \cdots & 0 \\ \binom{1}{0} & \binom{1}{1} & 0 & & & \vdots \\ \vdots & & \ddots & \ddots & & \vdots \\ \vdots & & & \ddots & \ddots & \vdots \\ \binom{m-1}{0} & & & & \binom{m-1}{m-1} & 0 \\ \binom{m}{0} & \cdots & \cdots & \cdots & \cdots & \binom{m}{m} \end{array}\right) \in \mathscr{M}_{m+1}(\mathbb{R}).$$
Deduce that for all $\left(u_{0}, \ldots, u_{m}\right), \left(v_{0}, \ldots, v_{m}\right) \in \mathbb{R}^{m+1}$,
$$\text{if} \quad \forall k \leqslant m, \quad u_{k} = \sum_{\ell=0}^{k} \binom{k}{\ell} v_{\ell}, \quad \text{then} \quad \forall k \leqslant m, \quad v_{k} = \sum_{\ell=0}^{k} (-1)^{k-\ell} \binom{k}{\ell} u_{\ell}$$