This chapter is the fulcrum against which the force of linear algebra is magnified into a powerful tool for understanding chain maps. The Escolar-Hiraoka decomposition theorem enables us to reason about chain maps in terms of their canonical decomposition into 10 different kinds of simpler chain maps. Not only does this dramatically simplify proofs of useful properties, it completely characterizes chain maps. With the Escolar-Hiraoka decomposition in hand, we learn a few surprising facts about chain complexes of vector spaces. There are two—apparently different—equivalences between chain complexes, quasi-isomorphism and chain homotopy equivalence. While these two equivalences are quite different in general for chain complexes of vector spaces, they agree. In a roundabout way, these two equivalences are also related to exact sequences of sequences. If one starts with a three-term exact sequence of chain complexes (instead of vector spaces), one can derive a much longer exact sequence involving the homology of these three chain complexes, a fact called the Snake Lemma.