Coordinate space

In mathematics, specifically in linear algebra, the coordinate space, Fⁿ, is the prototypical example of an n-dimensional vector space over a field F. It can be defined as the product space of F over a finite index set.

Definition

Let F denote an arbitrary field (such as the real numbers R or the complex numbers C). For any positive integer n, the space of all n-tuples of elements of F forms an n-dimensional vector space over F called coordinate space and denoted Fⁿ.

An element of Fⁿ is written

$\mathbf x = (x_1, x_2, \cdots, x_n)$

where each x_i is an element of F. The operations on Fⁿ are defined by

$\mathbf x + \mathbf y = (x_1 + y_1, x_2 + y_2, \cdots, x_n + y_n)$

$\alpha \mathbf x = (\alpha x_1, \alpha x_2, \cdots, \alpha x_n).$

The zero vector is given by

$\mathbf 0 = (0, 0, \cdots, 0)$

and the additive inverse of the vector x is given by

$-\mathbf x = (-x_1, -x_2, \cdots, -x_n).$

Matrix notation

In standard matrix notation, each element of Fⁿ is typically written as a column vector

$\mathbf x = \begin{bmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{bmatrix}$

and sometimes as a row vector:

$\mathbf x = \begin{bmatrix} x_1 & x_2 & \dots & x_n \end{bmatrix}.$

The coordinate space Fⁿ may then be interpreted as the space of all n×1 column vectors, or all 1×n row vectors with the ordinary matrix operations of addition and scalar multiplication.

Linear transformations from Fⁿ to F^m may then be written as m×n matrices which act on the elements of Fⁿ via left multiplication (when the elements of Fⁿ are column vectors) or right multiplication (when they are row vectors).

Standard basis

The coordinate space Fⁿ comes with a standard basis:

$\mathbf e_1 = (1, 0, \ldots, 0)$

$\mathbf e_2 = (0, 1, \ldots, 0)$

$\vdots$

$\mathbf e_n = (0, 0, \ldots, 1)$

where 1 denotes the multiplicative identity in F. To see that this is a basis, note that an arbitrary vector in Fⁿ can be written uniquely in the form

$\mathbf x = \sum_{i=1}^n x_i \mathbf{e}_i.$

Discussion

It is a standard fact of linear algebra that every n-dimensional vector space V over F is isomorphic to Fⁿ. It is a crucial point, however, that this isomorphism is not canonical. If it were, mathematicians would work only with Fⁿ rather than with abstract vector spaces.

A choice of isomorphism is equivalent to a choice of ordered basis for V. To see this, let

A : Fⁿ → V

be a linear isomorphism. Define an ordered basis {a_i} for V by

a_i = A(e_i) for 1 ≤ i ≤ n.

Conversely, given any ordered basis {a_i} for V define a linear map A : Fⁿ → V by

$A(\mathbf x) = \sum_{i=1}^n x_i \mathbf a_i.$

It is not hard to check that A is an isomorphism. Thus ordered bases for V are in 1-1 correspondence with linear isomorphisms Fⁿ → V.

The reason for working with abstract vector spaces instead of Fⁿ is that it is often preferable to work in a coordinate-free manner, i.e. without choosing a preferred basis. Indeed, many vector spaces that naturally show up in mathematics do not come with a preferred choice of basis.

It is possible and sometimes desirable to view a coordinate space dually as the set of F-valued functions on a finite set; that is, each "point" of Fⁿ is viewed as a function whose domain is the finite set {1,2....n} and codomain F. The function sends an element i of {1,2....n} to the value of the i'th coordinate of the "point", so Fⁿ is, dually, a set of functions.

See also

• real coordinate space, Rⁿ
• complex coordinate space, Cⁿ
• examples of vector spaces