Vector Basis — from Wolfram MathWorld
Excerpt
A vector basis of a vector space V is defined as a subset v_1,…,v_n of vectors in V that are linearly independent and span V. Consequently, if (v_1,v_2,…,v_n) is a list of vectors in V, then these vectors form a vector basis if and only if every v in V can be uniquely written as v=a_1v_1+a_2v_2+…+a_nv_n, (1) where a_1, …, a_n are elements of the base field. When the base field is the reals so that a_i in R for i=1,…,n, the resulting basis vectors are n-tuples of reals that span…
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],](https://mathworld.wolfram.com/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^^].](https://mathworld.wolfram.com/images/equations/VectorBasis/NumberedEquation5.svg){kind=small} | (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.
See also
Basis Vector, Bilinear Basis, Change of Basis, Dimension, Hilbert Basis, Linear Combination,Modular System Basis, Orthonormal Basis, Standard Basis, Topological Basis, Vector, Vector Space, Vector Space Span
Portions of this entry contributed by Todd Rowland
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Cite this as:
Rowland, Todd and Weisstein, Eric W. “Vector Basis.” From MathWorld—A Wolfram Web Resource. https://mathworld.wolfram.com/VectorBasis.html