Question 4.8.6: A Transition Matrix for Rotation in R² Find the coordinates ...
A Transition Matrix for Rotation in R²
Find the coordinates of a point (x, y)in R² relative to the basis
B´ = {(cos θ, sin θ), (-sin θ, cos θ)}.
By Theorem 4.21 you have
[B´ B] = \left [ \begin{matrix} cos θ& -sin θ& 1 & 0 \\ sin θ& cos θ& 0 & 1 \end{matrix} \right ] .
Because B is the standard basis for R², P^{-1} is represented by (B´)^{-1}. You can use the formula given in Section 2.3 (page 66) for the inverse of a 2 × 2 matrix to find (B´)^{-1}.
This results in
[I P^{-1}] = \left [ \begin{matrix} 1& 0& cos θ & sin θ \\ 0& 1& -sin θ & cos θ \end{matrix} \right ] .
By letting (x´, y´) be the coordinates of (x, y) relative to B´, you can use the transition matrix P^{-1} as follows.
\left [ \begin{matrix} cos θ& sin θ \\ -sin θ& cos θ \end{matrix} \right ]\left [ \begin{matrix} x \\ y \end{matrix} \right ] = \left [ \begin{matrix} x´ \\ y´ \end{matrix} \right ]
The x´ – and y´ – coordinates are x´ =x cos θ + y sin θ and y´ = -x sin θ + y cos θ.