Let $\{x_1, \dots, x_n\}$ be an orthonormal set of $\mathbb R^n$. Let $V$ be an arbitrary one-dimensional subspace. I am trying to determine the quantity \begin{align*} \sum_{j=1}^n \langle P_V(x_j), P_V(x_j) \rangle, \end{align*} where $P_V(x_j)$ denotes the orthogonal projection onto $V$.
Here is my strategy: Let $v \in V$ be a unit vector. We note \begin{align*} \langle P_V(x_j), P_V(x_j) \rangle = \langle (\langle x_j, v\rangle v), (\langle x_j, v\rangle v) \rangle = |\langle x_j, v \rangle |^2. \end{align*} So \begin{align*} \sum_1^n\langle P_V(x_j), P_V(x_j) \rangle = \sum_1^n |\langle x_j, v \rangle |^2 = \|v\|_2^2 = 1. \end{align*} Is this correct?