= Solution
The matrix has the standard <rotation matrix> form $\begin{pmatrix}\cos\theta&-\sin\theta\\\sin\theta&\cos\theta\end{pmatrix}$, with $\cos\theta=\sin\theta=1/\sqrt2$. Thus
$$
\boxed{\theta=\frac\pi4\pmod{2\pi}.}
$$
With $\omega=U\omega'$, 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.
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