Math.SE! I'd like some help understanding the premise of the following question:
A rotation of $\mathbb{R}^2$ about the origin is a linear mapping $R_\psi$ given by
$R_\psi$ $\begin{pmatrix} r\cos\phi \\ r\sin\phi \\ \end{pmatrix}$ = $\begin{pmatrix} r\cos(\phi+\psi) \\ r\sin(\phi+\psi) \\ \end{pmatrix}$
for $0\leq\psi<2\pi$ and where any vector $v\in \mathbb{R}^2$ can be written as $\begin{pmatrix} r\cos\phi \\ r\sin\phi \\ \end{pmatrix}$ where $r$ is the length of $v$ and $\phi$ is the angle between $v$ and the positive $x$-axis. Verify that $R_\psi = T_A$ where $A=[R_\psi]_E=\begin{pmatrix}\cos \ \psi&-\sin \ \psi\\ \sin \ \psi&\cos\ \psi\\ \end{pmatrix}$ and $T_A(v)=Av$ for $v \in V$.
It wasn't difficult to actually verify this result - my real question is this: how can I obtain the fact $[R_\psi]_E=\begin{pmatrix}\cos \ \psi&-\sin \ \psi\\ \sin \ \psi&\cos\ \psi\\ \end{pmatrix}$ (where $E$ is the standard basis), and, more generally, how do I determine what $[R_\psi]_B$ is for any arbitary basis $B$ of $\mathbb{R}^2$?
Thanks in advance for any help!