# Coordinate space

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{{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} 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

${\displaystyle \mathbf {x} =(x_{1},x_{2},\cdots ,x_{n})}$

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

${\displaystyle \mathbf {x} +\mathbf {y} =(x_{1}+y_{1},x_{2}+y_{2},\cdots ,x_{n}+y_{n})}$
${\displaystyle \alpha \mathbf {x} =(\alpha x_{1},\alpha x_{2},\cdots ,\alpha x_{n}).}$

The zero vector is given by

${\displaystyle \mathbf {0} =(0,0,\cdots ,0)}$

and the additive inverse of the vector x is given by

${\displaystyle -\mathbf {x} =(-x_{1},-x_{2},\cdots ,-x_{n}).}$

### Matrix notation

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

${\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}\\x_{2}\\\vdots \\x_{n}\end{bmatrix}}}$

and sometimes as a row vector:

${\displaystyle \mathbf {x} ={\begin{bmatrix}x_{1}&x_{2}&\dots &x_{n}\end{bmatrix}}.}$

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 Fm 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:

${\displaystyle \mathbf {e} _{1}=(1,0,\ldots ,0)}$
${\displaystyle \mathbf {e} _{2}=(0,1,\ldots ,0)}$
${\displaystyle \vdots }$
${\displaystyle \mathbf {e} _{n}=(0,0,\ldots ,1)}$

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

${\displaystyle \mathbf {x} =\sum _{i=1}^{n}x_{i}\mathbf {e} _{i}.}$

## 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 : FnV

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

ai = A(ei) for 1 ≤ in.

Conversely, given any ordered basis {ai} for V define a linear map A : FnV by

${\displaystyle A(\mathbf {x} )=\sum _{i=1}^{n}x_{i}\mathbf {a} _{i}.}$

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

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.