Coordinate space

From Wikipedia, the free encyclopedia

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.

1 Definition
- 1.1 Matrix notation
2 Standard basis
3 Discussion
4 See also

[edit] 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

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

[edit] Matrix notation

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

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

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.

[edit] 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

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

[edit] 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.