---- > [!definition] Definition. ([[reflection]]) > A **reflection** is a [[euclidean isometry]] $r: \mathbb{R}^{n} \to \mathbb{R}^{n}$ whose set of fixed points form a [[hyperplane]] in $\mathbb{R}^{n}$. > > If $E$ is a Euclidean space with [[inner product]] $(-,-)$, and $\lambda,\alpha \in E$ with $\alpha \neq 0$, define the **check-pairing**[^1] $\langle \lambda, \check{\alpha} \rangle: = \frac{2(\lambda, \alpha)}{(\alpha, \alpha)}.$ > The reflection $w_{\alpha}: E \to E$ across/along the hyperplane $H_{\alpha}=\text{span}^{\perp}(\alpha)=\{ x \in E: (x, \alpha)=0 \}$ is given by $\lambda \xmapsto{w_{\alpha}} \lambda - \langle \lambda, \check{\alpha} \rangle \alpha. $ > Easy to check that $x \in H_{\alpha} \implies w_{\alpha}(x)=x$ and $w_{\alpha}(\alpha)=-\alpha$, so this is indeed a reflection > > - [ ] ^ check that (in written notes) > - [ ] picture from written notes > > [!basicexample] Reflection across a line. > A reflection $r:\mathbb{R}^{2} \to \mathbb{R}^{2}$ across a [[line]] $\ell$, called an **axis of reflection**, is that transformation that leaves every point of $\ell$ fixed and takes every point $q$ not in $\ell$ to the point $q'$ such that $\ell$ is the perpendicular bisector of the line segment from $q$ to $q'$ (draw a picture). [^1]: Warning: this pairing is linear in its first argument, but not its second. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```