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I am reading about Dihedral Groups and I have following questions:

  • Elements of $D_n$ act as linear transformations of plane.

My thought:I know that $D_n=\{\langle a,b\rangle :a^n=b^2=1,bab=a^{-1}\}$ which comprises of rotations and reflections of the n-gon.

But then How to prove that rotations and reflections are linear transformations of plane?

  • Matrices for elements of $D_n$ have the form: $r_k$=\begin{bmatrix} \cos{\frac{2\pi k}{n}}&-\sin{\frac{2\pi k}{n}}\\\sin{\frac{2\pi k}{n}}&\cos{\frac{2\pi k}{n}}\end{bmatrix}

which is obtained by rotation of a n-gon by $\frac{2k\pi}{n}$.

and $s_k$=\begin{bmatrix} \cos{\frac{2\pi k}{n}}&\sin{\frac{2\pi k}{n}}\\\sin{\frac{2\pi k}{n}}&-\cos{\frac{2\pi k}{n}}\end{bmatrix}

is a reflection about a line which makes an angle $\frac{k\pi}{n}$ with x-axis.

My thought:

I know that rotation matrix is given by \begin{bmatrix} \cos{\theta}&-\sin{\theta}\\\sin{\theta}&\cos{\theta}\end{bmatrix}.

But I dont know how a reflection matrix looks like? How can I prove that the elements of $D_n$ can be represented like this?

NOTE: The two lines have been adopted from Wikipedia. But I need a proof of these facts which I cant prove using my knowledge.

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"How to prove that rotations and reflections are linear transformations of plane ?"

Rotations and reflections preserve straight lines in the plane - each line is mapped to another line by each rotation or reflection. Therefore rotations and reflections are linear transformations.

Another way to see this is to note that rotations and reflections are isometries - they preserve distances between points. And isometries are a subset of linear transformations.

There is a Wikipedia article on rotations and reflections in the plane: https://en.wikipedia.org/wiki/Rotations_and_reflections_in_two_dimensions

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