Coordinate Space - Discussion

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

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; 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.