# Coordinate space

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In mathematics, specifically in linear algebra, the **coordinate space**, **F**^{n}, 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 **F**^{n}.

An element of **F**^{n} is written

where each *x*_{i} is an element of **F**. The operations on **F**^{n} 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, each element of **F**^{n} is typically written as a column vector

and sometimes as a row vector:

The coordinate space **F**^{n} 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 **F**^{n} to **F**^{m} may then be written as *m*×*n* matrices which act on the elements of **F**^{n} via left multiplication (when the elements of **F**^{n} are column vectors) or right multiplication (when they are row vectors).

## Standard basis

The coordinate space **F**^{n} 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 **F**^{n} can be written uniquely in the form

## Discussion

It is a standard fact of linear algebra that every *n*-dimensional vector space *V* over **F** is isomorphic to **F**^{n}. It is a crucial point, however, that this isomorphism is not canonical. If it were, mathematicians would work only with **F**^{n} 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*:**F**^{n}→*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* : **F**^{n} → *V* by

It is not hard to check that *A* is an isomorphism. Thus ordered bases for *V* are in 1-1 correspondence with linear isomorphisms **F**^{n} → *V*.

The reason for working with abstract vector spaces instead of **F**^{n} 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 **F**^{n} 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 **F**^{n} is, dually, a set of functions.