Our study of homological algebra begins in earnest in this chapter, by studying sequences of linear maps. In this chapter, we prove the barcode decomposition theorem, which is the first of two theorems for sequences that are analogous to the dimension theorem. In the latter half of the chapter, we turn our attention to chain complexes, which are sequences that play a role in many homological applications. Our study of chain complexes is preparation for the Escolar-Hiraoka decomposition theorem, proven in the next chapter. Along the way, we develop algorithms for computing the barcode decomposition and its properties.