We have been adding vectors and scaling them for a while now. But when we do those operations, we usually stay inside some familiar set, like the plane or 3‑D space. This chapter makes that idea exact: we define what a vector space really is, and then look at smaller worlds inside it called subspaces. We will meet two of the most important subspaces that come from a matrix: the column space and the nullspace. They show the geometry behind linear systems.