1

How would you prove the following proposition...

The operations $r_h$ and $r_v$ commute, that is, $r_hr_v=r_vr_h$

where $r_h$ is a horizontal reflection and $r_v$ is a horizontal reflection.

I can see it is true if you were to draw out say a pattern and then do the reflections on it as you would end up with the same look, but I don't understand how I would write that out mathematical. Could someone demonstrate?

user6259845
  • 913
  • 1
    If you are working in just $\mathbb{R}^{2}$, there are matrices corresponding to a vertical reflection and to a horizontal reflection (which correspond to however you would define $r_{h}$ and $r_{v}$ for vectors). Proving that these commute would do it. – Bilbottom May 13 '18 at 21:36

1 Answers1

1

First of all, I think a bit more context is needed: you didn't specify reflections about what.

One way to tackle this question to introduce coordinates and develop formulas for these reflections. (Of course, I'm not saying that this is the only way, nor am I saying that this is the best way, but it is a way.)

For example, let's say $r_h$ is the reflection about the $x$-axis and $r_v$ is the reflection about the $y$-axis on the Cartesian plane. Then for any point $(x,y)$: $$r_h(x,y)=(x,-y) \quad \text{and} \quad r_v(x,y)=(-x,y).$$ Now you can apply those compositions to an arbitrary point in the plane. One of them is: $$r_hr_v(x,y)=r_h(r_v(x,y))=r_h(-x,y)=(-x,-y).$$ Similarly, you can calculate $r_vr_h(x,y)$.

zipirovich
  • 14,670
  • 1
  • 26
  • 35