Show $\phi: D_{2n} \to \mathrm{GL}_2(\mathbb{R})$ is an injective homomorphism, where $\phi: \textrm{rotations} \to \left( \begin{array}{cc} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array} \right) \\ \phi : \textrm{reflections} \to \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right)$
So $\mathrm{GL}_2(\mathbb{R})$ is the group of invertible $2 \times 2$ matrices.
For the homomorphism part, I have to show that $\phi(ab) = \phi(a) \phi(b)$ where $a$ and $b$ can be a combination of rotations, reflections, or both. So the matrix represented by the combination of the rotations/reflections is equal to the product of the matrices representing each of those rotations or reflections individually.
For the injective part, I need to show that if two matrices in $\mathrm{GL}_2(\mathbb{R})$ are equal, then the rotation/reflection that the two equal matrices represent are the same (i.e. $\phi(a) = \phi(b) \implies a = b$) (or the contrapositive).
So for the homomorphism part, do I need to do separate cases? i.e. where $a$ and $b$ are both rotations, both reflections, or a combination of both? Or is there a more general way to cover all the cases with a shorter proof? The same question goes for the injection; do I have to check both cases where $a = b$ are either both rotations or reflections, or is there an easier method? Is the fact that the matrices are invertible important, or relevant at all to our interests here?
Hint:
$D_{2n} = \langle r,s|r^n=s^2=1, rs= sr^{-1} \rangle$.
So a map $f$ will extend to a homomorphism if $f(r)^n =1, f(s)^2=1, f(r)f(s)=f(s)f(r)^{-1}$