Euclidean Space - Real Coordinate Space

Real Coordinate Space

Let R denote the field of real numbers. For any positive integer n, the set of all n-tuples of real numbers forms an n-dimensional vector space over R, which is denoted Rn and sometimes called real coordinate space. An element of Rn is written

where each x_i is a real number. The vector space operations on Rn are defined by

The vector space Rn comes with a standard basis:

$\begin{align} \mathbf{e}_1 & = (1, 0, \ldots, 0), \\ \mathbf{e}_2 & = (0, 1, \ldots, 0), \\ & {}\,\,\, \vdots \\ \mathbf{e}_n & = (0, 0, \ldots, 1). \end{align}$

An arbitrary vector in Rn can then be written in the form

Rn is the prototypical example of a real n-dimensional vector space. In fact, every real n-dimensional vector space V is isomorphic to Rn. This isomorphism is not canonical, however. A choice of isomorphism is equivalent to a choice of basis for V (by looking at the image of the standard basis for Rn in V). The reason for working with arbitrary vector spaces instead of Rn is that it is often preferable to work in a coordinate-free manner (that is, without choosing a preferred basis).