When given a coordinate system A:
\begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix}
a coordinate system rotated 90 degrees counterclockwise written in terms of A as $^A_BR$:
\begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix}
and a point $^BP$ written in terms of B:
\begin{bmatrix} 1 \\ -1 \end{bmatrix}
We can get $^AP$ by through the following equation:
$^AP = (^A_BR) (^BP)$
such that $^AP$ is:
\begin{bmatrix} 1 \\ 1 \end{bmatrix}
I've never found a formal proof that states why this is true for any point. It's slightly confusing to me why this works, and I feel like a formal proof might help me understand it better. I'm currently reading through Introduction to Robotics: Mechanics and Control, but they never gave an adequate description why this works. Any help?