Math 511: Linear Algebra

Reflections in $\mathbb{R}^2$, Part 1

Reflections on the Plane ¶

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In this project we will use transition matrices to determine the standard matrix representation of a reflection $L$ with respect to the line $ax + by = 0$.

1. ¶

Find the standard matrix representation for $L$, the reflection with respect to the line $x=0$.

2. ¶

Find the standard matrix representation for $L$, the reflection with respect to the line $y=0$.

3. ¶

Find the standard matrix representation for $L$, the reflection with respect to the line $x-y=0$.

4. ¶

For the linear transformation $\ell$ 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 $(0,5)$.

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