I was reading the following page from the Physics from Symmetry by Jakop Schwichtenberg, I didn't quite understand the calculation.
Basically what he does is the following:
$J_1$ is one of the basis element for the generators of the group $SO(3)$ and is given by \begin{align} J_1 = \begin{pmatrix} 0 & 0 & 0\\ 0 & 0 & -1\\ 0 & 1 & 0 \end{pmatrix} \end{align} He is trying to show that, $e^{\theta J_{1}}$ will produce a well known rotation matrix in 3D as in the bottom of the page. So far so good. What I don't understand is that he defines the lower right $2\times 2$ matrix $j_1$ in $J_1$ and uses this $2\times 2$ matrix to show that
\begin{align} e^{\theta j_{1}} = \begin{pmatrix} cos(\theta) & -sin(\theta)\\ sin(\theta) & cos (\theta) \end{pmatrix} \end{align}
But how does this helps us? How does he conclude the final result from it?
