We consider $n$ a strictly positive integer and $\mathscr{E}_{d}^{n}(\mathbb{R}) = \{ \boldsymbol{z} = (\boldsymbol{z}_{i})_{1 \leqslant i \leqslant n} \mid \boldsymbol{z}_{i} \in \mathbb{R}^{d}, 1 \leqslant i \leqslant n \}$. For all $\boldsymbol{x} \in \mathscr{E}_{d}^{n}(\mathbb{R})$, we denote $c(\boldsymbol{x}) = \{ \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R}) \mid \exists g \in \operatorname{Dep}(\mathbb{R}^{d}), g \cdot \boldsymbol{x} = \boldsymbol{y} \}$.
[(a)] Show that if $c(\boldsymbol{x}) \cap c(\boldsymbol{y}) \neq \emptyset$ then $c(\boldsymbol{x}) = c(\boldsymbol{y})$.
[(b)] Show that if $c(\boldsymbol{x}) = c(\boldsymbol{y})$ then $\delta(\boldsymbol{x}, \boldsymbol{y}) = 0$.
We consider $n$ a strictly positive integer and $\mathscr{E}_{d}^{n}(\mathbb{R}) = \{ \boldsymbol{z} = (\boldsymbol{z}_{i})_{1 \leqslant i \leqslant n} \mid \boldsymbol{z}_{i} \in \mathbb{R}^{d}, 1 \leqslant i \leqslant n \}$. For all $\boldsymbol{x} \in \mathscr{E}_{d}^{n}(\mathbb{R})$, we denote $c(\boldsymbol{x}) = \{ \boldsymbol{y} \in \mathscr{E}_{d}^{n}(\mathbb{R}) \mid \exists g \in \operatorname{Dep}(\mathbb{R}^{d}), g \cdot \boldsymbol{x} = \boldsymbol{y} \}$.
\begin{itemize}
\item[(a)] Show that if $c(\boldsymbol{x}) \cap c(\boldsymbol{y}) \neq \emptyset$ then $c(\boldsymbol{x}) = c(\boldsymbol{y})$.
\item[(b)] Show that if $c(\boldsymbol{x}) = c(\boldsymbol{y})$ then $\delta(\boldsymbol{x}, \boldsymbol{y}) = 0$.
\end{itemize}