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Span
The span of a set of vectors
of a vector space
over a field
is the set of linear combinations
with 
. It is denoted
. More generally, the span of a set 
(not necessarily finite) of vectors is the collection
of all (finite) linear combinations of elements of 
. The span of the empty set
is defined to be the singleton
consisting of the zero vector
.
For example, the standard basis vectors
and 
span
because every vector of
can be represented as a linear combination of 
and 
.
is a subspace
of 
and is the smallest subspace containing
.
Span is both a noun and a verb; a set of vectors can span a vector space, and a vector can be in the span of a set of vectors.
Checking span: To see whether a vector is in the span of other vectors, one can set up an augmented matrix, since if
is in the span of
, then
. This is a system of linear equations. Thus, if it has a solution,
is in the span of
. Note that the solution does not have to be unique for
to be in the span.
To see whether a set of vectors spans a vector space, you need to check that there are at least as many linearly independent
vectors as the dimension
of the space. For example, it can be shown that in
, 
vectors are never linearly independent, and 
vectors never span.
Remark. We can define the concept of span also for a module
over a ring
.
Given a subset
we define the module generated by
as the set of all finite linear combinations of elements of 
.
Be aware that in general there does not exist a linearly independent subset which generates
the whole module, i.e. there does not have to exist a basis.
Also, even
if 
is generated by 
elements, it is in general not true that any other set of 
linearly independent elements of 
spans 
.
For example
is generated by 
as a
-module but not by 
.