Let the random variables \(U_{1}\) and \(U_{2}\) be jointly Gaussian, with zero means, equal variances, and correlation coefficient \(ho eq 0\). Consider the new random variables \(V_{1}\) and \(V_{2}\) defined by a rotational transformation about the origin of the \((u, v)\) plane,

\[ \left[\begin{array}{l} v_{1} \\ v_{2} \end{array}\right]=\left[\begin{array}{rr} \cos \phi & \sin \phi \\ -\sin \phi & \cos \phi \end{array}\right]\left[\begin{array}{l} u_{1} \\ u_{2} \end{array}\right] \]

where \(\phi\) is the rotation angle. Show that if \(\phi\) is chosen to be \(45^{\circ}, V_{1}\) and \(V_{2}\) are independent random variables. What are the means and variances of \(V_{1}\) and \(V_{2}\) in this case?