This first chapter introduces vector space quotients, and develops both theoretical properties and computational algorithms for working with them. Special emphasis will be placed on effective algorithms for computing the quotient of two vector spaces, because this task will be done many times in the book. In the latter chapters, vector space quotients allow the interactions between linear maps in a sequence to be disentangled. While the first few sections are likely to be review material, they are essential for setting the stage since they establish basic notation and computational conventions that will be used throughout the book. The centerpiece of this chapter is the dimension theorem, which characterizes a single linear map and is proven with a constructive, algorithmic proof.