In mathematics, specifically in linear algebra, the coordinate space, Fⁿ, is the prototypical example of an n-dimensional vector space over a field F.

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

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

The zero vector is given by

and the additive inverse of the vector x is given by

Matrix notation

In standard matrix notation the elements of Fⁿ are written as column vectors

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

Linear transformations from F^m to Fⁿ may then be written as m×n matrices which act via left multiplication on the elements of Fⁿ.

In a similiar manner, the elements of the dual space (Fⁿ)* are written as row vectors, so the dual space may be interpretted as the space of all 1×n row vectors.

Standard basis

The coordinate space Fⁿ comes with a standard basis:

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

See also

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