I have learnt about the group
$$
SO(2) = \left\{\pmatrix{\cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta)} : \theta \in \mathbb{R} \right\}.
$$
This group has do with rotations. That is, if one multiplies a matrix from $SO(2)$ by a vector in $\mathbb{R}^2$ the result is the rotation of the vector by an angle of $\theta$.
I have been told that the $S$ in $SO(2)$ means that the matrices have determinant $1$. I have also been told that it makes sense to talk about $O(2)$. My question is: What exactly is $O(2)$? I can see that it has to be a bigger group, but is it just matrices from $SO(2)$ multiplied by constants?
(This is not homework, I am just interested.)
Best Answer
Every member of $O(2)$ must have determinant $\pm 1$. A particular example with determinant $-1$ is $\pmatrix{-1 & 0\cr 0 & 1\cr}$, let's call it $B$. Every member of $O(2)$ is either a member of $SO(2)$ or $B$ times a member of $SO(2)$, thus of the form $$ \pmatrix{-\cos(\theta) & \sin(\theta)\cr \sin(\theta) & \cos(\theta)\cr}$$