= Solution
For $N>1$, use the <orthonormal basis> $(|a\rangle,|\omega\rangle)$ from the preceding part. Both <reflection operators> preserve $\mathcal V$, because their reflected vectors belong to it. With $s=\sin\theta$ and $c=\cos\theta$, their restricted <matrices> are
$$
V_{|a\rangle}=\begin{pmatrix}-1&0\\0&1\end{pmatrix},\qquad
V_{|\psi_A\rangle}=I-2\begin{pmatrix}s^2&sc\\sc&c^2\end{pmatrix}
=\begin{pmatrix}\cos2\theta&-\sin2\theta\\-\sin2\theta&-\cos2\theta\end{pmatrix}.
$$
Multiplying in the specified order, including the overall minus sign, gives
$$
\boxed{G\big|_{\mathcal V}=\begin{pmatrix}\cos2\theta&\sin2\theta\\-\sin2\theta&\cos2\theta\end{pmatrix}.}
$$
Equivalently,
$$
G\big|_{\mathcal V}=\cos2\theta\bigl(|a\rangle\langle a|+|\omega\rangle\langle\omega|\bigr)
+\sin2\theta\bigl(|a\rangle\langle\omega|-|\omega\rangle\langle a|\bigr).
$$
In particular, $G$ sends $\sin t|a\rangle+\cos t|\omega\rangle$ to $\sin(t+2\theta)|a\rangle+\cos(t+2\theta)|\omega\rangle$. This proves the <Grover rotation angle> description with the signs appropriate to these <reflection operators>.
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