Chapter 2.3, Problem 29E

Consider the matrix $D_{\alpha }=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]$ .
We know that the linear transformation $T\left(\stackrel{\to }{x}\right)=D_{\alpha }\stackrel{\to }{x}$ isa counterclockwise rotation through an angle a.
a. For Iwo angles. $\alpha$ and $\beta$ , consider the products $\stackrel{\to }{y}=D_{\alpha }{D}_{\beta }\stackrel{\to }{x}$ and $\stackrel{\to }{y}=D_{\beta }{D}_{\alpha }\stackrel{\to }{x}$ . Arguing geometrically, describethe linear transformations $\stackrel{\to }{y}=D_{\alpha }{D}_{\beta }\stackrel{\to }{x}$ and $\stackrel{\to }{y}=D_{\beta }{D}_{\alpha }\stackrel{\to }{x}$ . Are the two transformations the same?
b. Now compute the products $D_{\alpha }{D}_{\beta }$ and $D_{\beta }{D}_{\alpha }$ . Dothe results make sense in terms of your answer inpart (a)? Recall the trigonometric identities
$\begin{array}{l}\sin \left(\alpha \pm \beta \right)=\sin \alpha \cos \beta \pm \cos \alpha \sin \beta \\ \sin \left(\alpha \pm \beta \right)=\cos \alpha \cos \beta \mp \sin \alpha \sin \beta \end{array}$