I need help with the following abstract algebra problem. It is not homework, but I need a solution.
Let $SL_2(\mathbb{R}) = \left\{ \big( \begin{smallmatrix} a & b\\ c & d\\\end{smallmatrix} \big)\;\middle\vert\;a,b,c,d \in \mathbb{R}, ad - bc = 1 \right\},$
$SO_2(\mathbb{R}) = \left\{ \big( \begin{smallmatrix} a & b\\ -b & a\\\end{smallmatrix} \big)\;\middle\vert\;a,b,c,d \in \mathbb{R}, a^2 + b^2 = 1 \right\}.$
Prove that $ SL_2(\mathbb{R})$ is a group, that $SO_2(\mathbb{R}) \leq SL_2(\mathbb{R})$, and that $SL_2(\mathbb{R})$ can be represented as a union of nonintersecting left classes with respect to $SO_2(\mathbb{R})$ in the form:
$SL_2(\mathbb{R}) = \bigcup_{\substack{r,p \in \mathbb{R}\\ r>0}} \begin{pmatrix} r & 0 \\ r^{-1}p & r^{-1} \\ \end{pmatrix} \cdot SO_2(\mathbb{R})$