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skygao

Question: Can someone explain what exactly is the difference between reflection about both x and y, and clockwise rotation about the origin by PI? I believe Keenan mentioned this in lecture and people are also having related problems on the assignment.

dvernet

Didn't we prove that they're equivalent in lecture? Consider the following where $k \in \mathbb{R}$:

$$\mathbf{A} = \begin{bmatrix} k & 0 \\ 0 & k \end{bmatrix}$$

$$\mathbf{B} = \begin{bmatrix} k\cos(\theta) & k-\sin(\theta) \\ k\sin(\theta) & k\cos(\theta) \end{bmatrix}$$

We know that reflection about both $x$ and $y$ for $\mathbf{A}$ is equivalent to multiplying $A$ by -1 (i.e. multiplying the $x$ and $y$ coordinates by -1):

$$\mathbf{A}_{reflect} = \begin{bmatrix} -k & 0 \\ 0 & -k \end{bmatrix}$$

Similarly, we know that reflection about both axes is equivalent to setting $\theta = \pi$ for $B$, giving us:

$$\mathbf{B}_{reflect} = \begin{bmatrix} k\cos(\pi) & -k\sin(\pi) \\ k\sin(\pi) & k\cos(\pi) \end{bmatrix} = \begin{bmatrix} -k & 0 \\ 0 & -k \end{bmatrix}$$

$$= \mathbf{A}_{reflect}$$

This proves that reflection is equivalent to rotation by $\pi$ in $\mathbb{R}^{2}$.

Also, is there a better way to typeset matrices in these comments (looks like we can't use the amsmath package). Also, sorry if you got email spammed while I was trying to edit this into proper latex.

skygao

@dvernet. Yes indeed. I guess I was really thinking of something else regarding the problem people are having on the assignment: let's say we have a unit-sized box with its bottom-left corner on the origin. What's the difference between reflecting it about the Y axis and translating it by (-1,0)?

kayvonf

@dvernet: use bmatrix.

 \mathbf{M} =
\begin{bmatrix}
a & b \\\
c & d
\end{bmatrix}


$$\mathbf{M} = \begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

Note: There does seem to be an issue where our Markdown parsing is stripping a slash, so notice the \\\ rather than the standard \\ latex newline.