Chapter 2.7, Problem 12E

A rotation in ℝ² usually requires four multiplications. Compute the product below, and show that the matrix for a rotation can be factored into three shear transformations (each of which requires only one multiplication).

$\left[\begin{array}{ccc}1& -\tan \phi /2& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ \sin \varphi & 1& 0\\ 0& 0& 1\end{array}\right]$

$\left[\begin{array}{ccc}1& -\tan \phi /2& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]$