For all $g = (\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$, we denote by $\phi_{g} : \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ the map defined by $\phi_{g}(x) = Rx + \tau$.
[(a)] Verify that for all $a, b \in \mathbb{R}^{d}$ and $g \in \operatorname{Dep}(\mathbb{R}^{d})$, we have $|\phi_{g}(a) - \phi_{g}(b)| = |a - b|$.
[(b)] Show that for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$, we have $\phi_{g} = \phi_{g^{\prime}}$ if and only if $g = g^{\prime}$.
[(c)] Show that there exists a unique $e \in \operatorname{Dep}(\mathbb{R}^{d})$ such that $\phi_{e}$ is the identity map on $\mathbb{R}^{d}$, that is $\phi_{e}(x) = x$ for all $x \in \mathbb{R}^{d}$.
For all $g = (\tau, R) \in \operatorname{Dep}(\mathbb{R}^{d})$, we denote by $\phi_{g} : \mathbb{R}^{d} \rightarrow \mathbb{R}^{d}$ the map defined by $\phi_{g}(x) = Rx + \tau$.
\begin{itemize}
\item[(a)] Verify that for all $a, b \in \mathbb{R}^{d}$ and $g \in \operatorname{Dep}(\mathbb{R}^{d})$, we have $|\phi_{g}(a) - \phi_{g}(b)| = |a - b|$.
\item[(b)] Show that for all $g, g^{\prime} \in \operatorname{Dep}(\mathbb{R}^{d})$, we have $\phi_{g} = \phi_{g^{\prime}}$ if and only if $g = g^{\prime}$.
\item[(c)] Show that there exists a unique $e \in \operatorname{Dep}(\mathbb{R}^{d})$ such that $\phi_{e}$ is the identity map on $\mathbb{R}^{d}$, that is $\phi_{e}(x) = x$ for all $x \in \mathbb{R}^{d}$.
\end{itemize}