What's the intuition behind the 2D rotation matrix? Can anyone offer an intuitive proof of why the 2D rotation matrix works?
http://en.wikipedia.org/wiki/Rotation_matrix
I've tried to derive it using polar coordinates to no avail.

If I rotate $(1,0)^T$ by an angle of $\theta$ counterclockwise, it should end up at $(\cos\theta,\sin\theta)^T$. This will be the first column in the rotation matrix.
If I rotate $(0,1)^T$ by an angle of $\theta$ counterclockwise, it should end up at $(-\sin\theta,\cos\theta)^T$. This will be the second column in the rotation matrix.
