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 .
|Date of creation||2013-03-22 11:58:18|
|Last modified on||2013-03-22 11:58:18|
|Last modified by||mathwizard (128)|