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
![[x_1; x_2; x_3; x_4; x_5]=x_2[-1; 1; 0; 0; 0]+x_3[-1; 0; 1; 0; 0]+x_4[-1; 0; 0; 1; 0]+x_5[-1; 0; 0; 0; 1],](/images/equations/VectorBasis/NumberedEquation4.svg) |
(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
![A^'=[Ax_1^^ ... Ax_n^^].](/images/equations/VectorBasis/NumberedEquation5.svg) |
(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.
