Past exam of the mathematical methods I course of the University of Oxford / 2017 / 8 / b / iii / Solution

The matrix has the standard rotation matrix form $\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)$, with $\cos \theta =\sin \theta =1/\sqrt{2}$. ThusWith $\omega =U{\omega }^{\prime }$, the new first axis points along the original $(1,1)$ direction and is counterclockwise from the original first axis. Specifying this coordinate convention avoids confusing rotation of the axes with the inverse transformation of component columns.