Euclidean Spaces of Many Dimensions

Abstract

In the preceding chapters, we have studied the main concepts concerning vectors in the familiar three-dimensional space. It was pointed out that the maximal number of linearly independent vectors was characteristic of those sets which could be identified pictorially with either a line, plane, or the entire space. Such a close association between the number of base vectors and the dimension of a space proves to be so fundamental that the generalization of the very idea of space proceeds virtually parallel to the extension of the idea of vector. Clearly, the simplest extension which comes to mind consists of imagining a vector in a space whose dimension is greater than three. It turns out that this can be done on various levels of abstraction, and we shall learn about such constructions in subsequent chapters. At this stage, we wish to give some thought to the notion of components, which was so characteristic of a vector in Euclidean three-space. With this in mind, we adopt the following definition.