A vector basis of a vector space is defined as a subset of vectors in that are linearly independent and span . Consequently, if is a list of vectors in , then these vectors form a vector basis if and only if every can be uniquely written as
(1)
|
where , ..., are elements of the base field.
When the base field is the reals so that for , the resulting basis vectors are -tuples of reals that span -dimensional Euclidean space . Other possible base fields include the complexes , as well as various fields of positive characteristic considered in algebra, number theory, and algebraic geometry.
A vector space has many different vector bases, but there are always the same number of basis vectors in each of them. The number of basis vectors in is called the dimension of . Every spanning list in a vector space can be reduced to a basis of the vector space.
The simplest example of a vector basis is the standard basis in Euclidean space , in which the basis vectors lie along each coordinate axis. A change of basis can be used to transform vectors (and operators) in a given basis to another.
Given a hyperplane defined by
(2)
|
a basis can be found by solving for in terms of , , , and . Carrying out this procedure,
(3)
|
so
(4)
|
and the above vectors form an (unnormalized) basis.
Given a matrix with an orthonormal basis, the matrix corresponding to a change of basis, expressed in terms of the original is
(5)
|
When a vector space is infinite dimensional, then a basis exists as long as one assumes the axiom of choice. A subset of the basis which is linearly independent and whose span is dense is called a complete set, and is similar to a basis. When is a Hilbert space, a complete set is called a Hilbert basis.