Let _ be the line spanned by a R2, and let R_ : R2 R2 be the linear map defined by

QUESTION:

Let \(\ell\) be the line spanned by \(\mathbf{a} \in \mathbb{R}^2\), and let \(R_{\ell}: \mathbb{R}^2 \rightarrow \mathbb{R}^2\) be the linear map defined by reflection across \(\ell\). Using the formula \(R_{\ell}(\mathbf{x})=\mathbf{x}^{\|}-\mathbf{x}^{\perp}\) given in Example 3, verify that

a. \(\left\|R_{\ell}(\mathbf{x})\right\|=\|\mathbf{x}\|\) for all \(\mathbf{x} \in \mathbb{R}^2\).

b. \(R_{\ell}(\mathbf{x}) \cdot \mathbf{a}=\mathbf{x} \cdot \mathbf{a}\) for all \(\mathbf{x} \in \mathbb{R}^2\); i.e., the angle between \(\mathbf{x}\) and \(\ell\) is the same as the angle between \(R_{\ell}(\mathbf{x})\) and \(\ell\).