Coordinate space

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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[edit]

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 xi 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).

Matrix notation[edit]

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

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

and sometimes as a row vector:

\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[edit]

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.

Discussion[edit]

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

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.

See also[edit]