Publisher Summary The purpose of this chapter is to begin the study of the theory of linear operators, which are basic to differential equations. This chapter presents an outline of the necessary facts about vector spaces. It is very useful to be able to treat vectors (and later, operators) as objects independent of any particular coordinate system. Henceforth this chapter shall use the term vector space to mean “subspace of a Cartesian space.” An element of a vector space will be called a vector (also a point). To distinguish them from vectors, real numbers are called scalar.