I am intending to learn Lie Algebras by following the online course on MIT OCW webpage.
In the preface section of the class, the following is stated in the lecture notes:
"Consider the set of isometrics of the plane $\mathbb{R}^2$. Clearly they form a group under composition, we denote this group by M(2). We can parametrize this group. If $g\in M(2)$, let $t$ be a translation such that g(0)=t(0). Then $t^{-1}g(0)=0$. Let $e_1,e_2$ denote the coordinate vectors in $(1,0)$ and $(0,1)$. Choose a rotation $k$ around the origin $0$ such that $k(e_1)=t^{-1}g(e_1)$. Then $h=k^{-1}t^{-1}g$ fixes both $0$ and $e_1$. "
Things are good so far but I do not understand the next sentence which states:
"Since $h(e_2)$ lies on the circles with centers 0 and $e_1$ we have either $h(e_2)=e_2$ or $rh(e_2)=e_2$ where $r$ is the reflection in the x-axis."
I do understand - by using the basic properties of isometries- $h(e_2)$ lies on the circles with centers 0 and $e_1$, but I do not understand rest. Actually the intersection of two circles is the points $(\dfrac{1}{\sqrt{2}},\dfrac{1}{\sqrt{2}})$ and $(-\dfrac{1}{\sqrt{2}},-\dfrac{1}{\sqrt{2}})$ , so I think they must be $h(e_2)$. Am I missing something?
Thank you for your help.
The vector $h(e_2)$ lies in the circle with center $0$ and radius $1$ and also in the circle with center $e_1$ and radius $\sqrt2$. These circles intersect at two and only two points: $\pm e_2$.