- #1

Kara386

- 208

- 2

## Homework Statement

Let ##R(\theta) = \left( \begin{array}{cc}

\cos(\theta) & -\sin(\theta)\\

\sin(\theta)& \cos(\theta)\\ \end{array} \right) \in O(2)## represent a rotation through angle ##\theta##, and

##X(\theta) = \left( \begin{array}{cc}

\cos(\theta) & \sin(\theta)\\

\sin(\theta)& -\cos(\theta)\\ \end{array} \right) \in O(2)## represent reflection around ##\frac{\theta}{2}##. Let m be a positive integer and H be the set such that ##H = \{R(\frac{2q\pi}{m}), X(\frac{2q\pi}{m}) | q = 0, 1, 2..., m-1\}##.

Calculate ##R(\theta)R(\phi)##, ##R(\theta)X(\phi)## and ##X(\theta)X(\phi)##, express answers in terms of R and X. Show H forms a subgroup of ##O(2)##.

## Homework Equations

## The Attempt at a Solution

##R(\theta) R(\phi) =

\left( \begin{array}{cc}

\cos(\theta+\phi) & -\sin(\theta+\phi)\\

\sin(\theta+\phi)& \cos(\theta+\phi)\\ \end{array} \right) = R(\theta + \phi)##

##R(\theta) X(\phi) =

\left( \begin{array}{cc}

\cos(\theta+\phi) & \sin(\theta+\phi)\\

\sin(\theta+\phi)& -\cos(\theta+\phi)\\ \end{array} \right) = X(\theta + \phi)##

##R(\theta) X(\phi) = R(\theta - \phi)##

I'm not really sure of the significance of these calculations, or if there is one. Does this have a geometric interpretation? I've been told that the set ##O(2)## is somehow related to the permutation of 3 points but haven't been able to find out why.

To show it's a subgroup of O(2) I have to show ##R \times X \in O(2)## where ##\times## is matrix multiplication and ##a^{-1} \in H##. Do I need to use the argument ##\frac{2q\pi}{m}## in these calculations? So for example let ##a = R(\frac{2q\pi}{m})##. Any help is very much appreciated, thank you! :)

I've gone ahead and attempted the problem with the argument above, and set a = R, b = X. I've already calculated RX to be ##= R(\theta - \phi)## above: ## =

\left( \begin{array}{cc}

\cos(0) & \sin(0)\\

\sin(0)& -\cos(0)\\ \end{array} \right)

=

\left( \begin{array}{cc}

1& 0\\

0 & 1\\ \end{array} \right)##

I wasn't really expecting to get the identity matrix, I suppose I show it's part of O(2) by showing ##A^T A = I ## and of course, it is a real 2x2 matrix.

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