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Vector Space Basis
A basis of a vector space 
is defined as
a subset 
of vectors in 
that are linearly
independent and vector space span 
. Consequently,
if 
is a list of vectors
in 
, then these vectors form a basis if
and only if every 
can be uniquely written as
 |
where 
, ..., 
are elements
of the base field. (Outside of pure mathematics, the base field is almost always
or 
, but fields of
positive characteristic are often considered in algebra, number theory, and algebraic
geometry). A vector space 
will have many
different 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 basis is the standard basis in 
consisting of
the coordinate axes. For example, in 
, the standard
basis consists of two vectors 
and 
. Any vector 
can be written uniquely as the linear
combination 
. Indeed, a vector is defined
by its coordinates. The vectors 
and 
are also a basis for 
because any
vector 
can be uniquely
written as 
. The above figure
shows 
, which are linear
combinations of the basis 
.
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.
Contact the MathWorld Team
ackermann[2,3]
