 Solution

Rotate the specified line onto the horizontal axis, stretch horizontally by two while leaving the orthogonal component unchanged, then rotate back. With $\theta =\pi /3$ and counterclockwise rotation matrix $R(\theta )$, the requested product isThe rightmost rotation acts first. In particular, $R(\theta )=\left(\begin{array}{cc}1/2& -\sqrt{3}/2\\ \sqrt{3}/2& 1/2\end{array}\right)$ and $R(-\theta )=R(\theta {)}^T$. For the unit direction $u=(1/2,\sqrt{3}/2{)}^T$, the same transformation is $T=I+u{u}^T$. It sends $u$ to $2u$ and fixes every orthogonal vector, verifying that the product is a directional stretch rather than a shear mapping or scaling in every direction. The constructed second panel in the geometric figure in Question 5 illustrates this separation.