Reflections on the Plane, Part 2¶
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In this project, we will use projections to determine the standard matrix representation of the reflection $L$ on the plane with respect to the line $ax + by = 0$. Recall that the projection of the vector $\mathbf{u}$ onto vector $\mathbf{v}$ is given by
$$ \text{proj}_{\mathbf{v}}\mathbf{u} = \frac{\mathbf{u}\cdot\mathbf{v}}{\mathbf{v}\cdot\mathbf{v}}\mathbf{v} $$
1.¶
Find the standard matrix representation for $L$, the projection onto the $y$-axis. That is find the standard matrix representation of the linear transformation on $\mathbb{R}^2$ defined by
$$ L(\mathbf{u}) = \text{proj}_{\jhat}\mathbf{u} $$
2.¶
Find the standard matrix representation the projection onto the $x$-axis.
3.¶
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 projection onto $\ell$ 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 projection onto $\ell$ 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 $\ell$.
4.¶
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 projection onto $\ell$ 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 projection onto $\ell$ with respect to the standard basis.
5.¶
Show that the projection
$$ \text{proj}_{\mathbf{v}}\mathbf{u} = \frac{1}{2}\left( \mathbf{u} + L(\mathbf{u}) \right), $$
where $L$ is the reflection with respect to the line $\ell = \text{Span}\left\{\mathbf{v}\right\}$. Using the standard matrix representation of the projection onto $\ell$, solve the equation for the standard matrix representation of $L$ and compare your result with the formula for $L$ found in the previous project.