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

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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

$\backslash mathbf\; x\; =\; (x\_1,\; x\_2,\; \backslash cdots,\; x\_n)$

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

$\backslash mathbf\; x\; +\; \backslash mathbf\; y\; =\; (x\_1\; +\; y\_1,\; x\_2\; +\; y\_2,\; \backslash cdots,\; x\_n\; +\; y\_n)$

$\backslash alpha\; \backslash mathbf\; x\; =\; (\backslash alpha\; x\_1,\; \backslash alpha\; x\_2,\; \backslash cdots,\; \backslash alpha\; x\_n)$

The zero vector is given by

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

and the additive inverse of the vector x is given by

$-\backslash mathbf\; x\; =\; (-x\_1,\; -x\_2,\; \backslash cdots,\; -x\_n)$

Matrix notation

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

$\backslash mathbf\; x\; =\; \backslash begin\{bmatrix\}\; x\_1\; \backslash \backslash \; x\_2\; \backslash \backslash \; \backslash vdots\; \backslash \backslash \; x\_n\; \backslash end\{bmatrix\}$

The coordinate space Fⁿ may then be interpreted as the space of all n×1 column 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 via left multiplication on the elements of Fⁿ.

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

Standard basis

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The coordinate space Fⁿ comes with a standard basis:

$\backslash mathbf\; e\_1\; =\; (1,\; 0,\; \backslash ldots,\; 0)$

$\backslash mathbf\; e\_2\; =\; (0,\; 1,\; \backslash ldots,\; 0)$

$\backslash vdots$

$\backslash mathbf\; e\_n\; =\; (0,\; 0,\; \backslash ldots,\; 1)$

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

$\backslash mathbf\; x\; =\; \backslash sum\_\{i=1\}^n\; x\_i\; \backslash mathbf\{e\}\_i$

Discussion

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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

$A(\backslash mathbf\; x)\; =\; \backslash sum\_\{i=1\}^n\; x\_i\; \backslash mathbf\; a\_i$

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.

See also

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

Linear algebra

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