In this chapter we introduce our first main protagonist — vectors — and present some of their fundamental properties. Throughout almost all of this text, we will only need to deal with vectors that have real entries, and this restriction will help to streamline and focus the exposition. Vectors with a prescribed number of entries fill out what is known as Euclidean space, since it forms the realm of ordinary Euclidean geometry. The entries of a vector can be viewed as Cartesian coordinates, and their number indicates the underlying dimension.