We consider $n$ a strictly positive integer and $\mathscr{E}_{d}^{n}(\mathbb{R}) = \{ \boldsymbol{z} = (\boldsymbol{z}_{i})_{1 \leqslant i \leqslant n} \mid \boldsymbol{z}_{i} \in \mathbb{R}^{d}, 1 \leqslant i \leqslant n \}$ the vector space of families of $n$ points in $\mathbb{R}^{d}$ equipped with the norm $\|\boldsymbol{z}\| = \sqrt{\sum_{i=1}^{n} |\boldsymbol{z}_{i}|^{2}}$. For all $g \in \operatorname{Dep}(\mathbb{R}^{d})$ and $\boldsymbol{z} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ we denote $g \cdot \boldsymbol{z} = (\phi_{g}(\boldsymbol{z}_{i}))_{1 \leqslant i \leqslant n}$.
\begin{itemize}
\item[(a)] Show that for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$ and $\boldsymbol{z} \in \mathscr{E}_{d}^{n}(\mathbb{R})$, we have $g \cdot (g^{\prime} \cdot \boldsymbol{z}) = (gg^{\prime}) \cdot \boldsymbol{z}$.
\item[(b)] Show that for all $\boldsymbol{x}, \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R})$ and all $g \in \operatorname{Dep}(\mathbb{R}^{d})$, if $\boldsymbol{x} = g \cdot \boldsymbol{y}$ then $\boldsymbol{y} = g^{-1} \cdot \boldsymbol{x}$.
\end{itemize}