Let $V$ and $W$ be subspaces of $\mathbb{R}^n$ and $P_V$ be orthogonal projection on $V$. The linear operator $T:V \to V$ is the restriction of $P_V\circ P_W$ to $V$. Prove that there is an orthonormal basis $ \{\vec{v_1} , \vec{v_2} , \dots , \vec{v_k}\}$ for $V$ of the eigenvectors operator $T$ and $ \{T(\vec{v_1}) , T(\vec{v_2}) , \dots , T(\vec{v_k})\}$ is orthogonal in $W$. Also how nonzero vectors of this set determine the angle between $V$ and $W$?
For the first part, I think we should use Principal axis theorem but I'm not sure. I don't understand the meaning of "angle" here and how can $\vec{v_i}$ be zero? If it's the eigenvector so is nonzero.