A vector basis of a finite-dimensional 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 iff every 
can be uniquely written as
 |
(1)
|
where 
,
..., 
are expansion
coefficients in the base field.
For an 
-dimensional real vector
space, a choice of vector basis identifies each vector
with an n-tuple of real
numbers, giving an isomorphism with 
. Other possible base fields include
the complex numbers 
, as well as various fields of positive
characteristic considered in algebra,
number theory, and algebraic
geometry.
A vector space 
can have 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 vector 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 vector basis to another.
Given a hyperplane defined by
 |
(2)
|
a vector 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) vector basis.
Let 
represent a linear
transformation in an old vector basis, and let the columns of 
be the coordinates of a new
vector basis in the old one. The matrix in the new vector
basis is
 |
(5)
|
When a vector space is infinite dimensional, a vector basis still exists assuming the axiom of choice; each vector is a finite linear combination of its elements. In a topological vector space, a set whose vector space span is dense is called total or complete. In a Hilbert space, a complete orthonormal basis is called a Hilbert basis. A complete linearly independent set need not be an orthonormal set, and hence need not be a Hilbert basis.
