Math 511: Linear Algebra
Reflections in $\mathbb{R}^2$, Part 1
Reflections on the Plane¶
$$ \require{color} \definecolor{brightblue}{rgb}{.267, .298, .812} \definecolor{darkblue}{rgb}{0.0, 0.0, 1.0} \definecolor{palepink}{rgb}{1, .73, .8} \definecolor{softmagenta}{rgb}{.99,.34,.86} \definecolor{blueviolet}{rgb}{.537,.192,.937} \definecolor{jonquil}{rgb}{.949,.792,.098} \definecolor{shockingpink}{rgb}{1, 0, .741} \definecolor{royalblue}{rgb}{0, .341, .914} \definecolor{alien}{rgb}{.529,.914,.067} \definecolor{crimson}{rgb}{1, .094, .271} \def\ihat{\mathbf{\hat{\unicode{x0131}}}} \def\jhat{\mathbf{\hat{\unicode{x0237}}}} \def\khat{\mathrm{\hat{k}}} \def\tombstone{\unicode{x220E}} \def\contradiction{\unicode{x2A33}} $$
In this project we will use transition matrices to determine the standard matrix representation of a reflection $L$ with respect to the line $ax + by = 0$.
3.¶
Find the standard matrix representation for $L$, the reflection with respect to the line $x-y=0$.
4.¶
For the line $\ell$ represented by the equation $x - 2y = 0$, find a vector $\mathbf{v}$ parallel to $\ell$, and a vector $\mathbf{w}$ perpendicular to $\ell$. Make your vectors $\mathbf{v}$ and $\mathbf{w}$ simple so that your computations are not overly complicated.
Determine the matrix $A_U$ for the reflection on $\ell$ with respect to the ordered basis $U = \left\{\mathbf{v},\mathbf{w}\right\}$.
Finally use the appropriate transition matrices to compute the standard matrix representation $A$ of $\ell$ with respect to the standard basis.
Use the standard matrix representation to find the reflections of the points $(2,1)$, $(-1,2)$, and $(5,0)$ with respect to $\ell$.
5.¶
Consider the line $\ell$ that is represented by the equation $ax + by = 0$. Find a vector $\mathbf{v}$ parallel to $\ell$ and a vector $\mathbf{w}$ that is perpendicular to $\ell$.
Determine the matrix $A_U$ for the reflection on $\ell$ with respect to the ordered basis $U = \left\{\mathbf{v},\mathbf{w}\right\}$.
Finally use the appropriate transition matrices to compute the standard matrix representation $A$ of the reflection on $\ell$ with respect to the standard basis.
6.¶
Find the standard matrix representation for the reflection with respect to the line $3x + 4y = 0$ using the formula you derived in problem 5. Use this matrix to find the images of the points $(3,4)$, $(-4,3)$, and $(0,5)$.