Here's a fact (in bold) I encountered in a proof that I'm having trouble justifying:
Given an orthogonal projector $P$, let it project onto $S_1$ along $S_2$. By definition those sets are orthogonal. Let $\{q_1, \dots , q_m\}$ be an orthonormal basis for $\mathbb{C}^m$, where $\{q_1, \dots, q_n\}$ is a basis for $S_1$ and $\{q_{n+1}, \dots , q_m\}$ is a basis for $S_1$...
How do I know that given an orthogonal projector that I can divide an orthonormal basis for the whole vector space into parts that form an orthonormal basis for the complementary subspaces formed by an orthogonal projector?
The bolded sentence doesn't mean you can divide any such basis this way. It means you can choose a particular basis using this strategy.
The strategy is to choose an arbitrary basis for $S_1$ and an arbitrary basis for $S_2$. Then combine them. Since $S_1$ and $S_2$ are orthogonal complements, the combined basis spans the whole space; since every vector in $S_1$ is orthogonal to every vector in $S_2$, and each individual basis is orthonormal, the combined basis is orthonormal.