On a pair of vector spaces

The paper provides an extensive investigation of a pair of vectors spaces, with a particular attention paid to perpendicular subspaces. Several new properties of the pair are identified, some of which enabled to derive known facts about subspaces by means of simpler proofs. The approach utilized exploits partitioned representations of orthogonal projectors and is based on the fact that there is one-to-one correspondence between an orthogonal projector and the subspace onto which it projects.