ABSTRACT
Using the familiar three-dimensional real vectors in IE→3 we can construct three-dimensional complex vectors explicitly, e.g., we define their algebraic operations and scalar product explicitly. Important properties can then be derived from all these explicit definitions. This constructive method provides an intuitive way to define things. However, such a method is not useful when we want to generalise our theory. Many of the explicit constructions cannot be easily generalised. When extending to higher dimensions what we want is to preserve the intrinsic and desirable properties of our three-dimensional complex vector space IE→3c, but not necessarily the constructive expressions. The idea is to define various quantities, including vector spaces themselves, by their properties from the start. This chapter starts by defining an N-dimensional complex vector space as a set of elements endowed with certain properties. There is no need to describe these elements explicitly as long they possess the desired properties.
