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 xi 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).

Read more about this topic:  Euclidean Space

Famous quotes containing the words real and/or space:

    “When much intercourse with a friend has supplied us with a standard of excellence, and has increased our respect for the resources of God who thus sends a real person to outgo our ideal; when he has, moreover, become an object of thought, and, whilst his character retains all its unconscious effect, is converted in the mind into solid and sweet wisdom,—it is a sign to us that his office is closing, and he is commonly withdrawn from our sight in a short time.”
    —Ralph Waldo Emerson (1803–1882)

    “What a phenomenon it has been—science fiction, space fiction—exploding out of nowhere, unexpectedly of course, as always happens when the human mind is being forced to expand; this time starwards, galaxy-wise, and who knows where next.”
    —Doris Lessing (b. 1919)