Linear span
In linear algebra, the linear span (also called the linear hull or just span) of a set of vectors in a vector space is the intersection of all subspaces containing that set. The linear span of a set of vectors is therefore a vector space. Spans can be generalized to matroids and modules.
Given a vector space V over a field K, the span of a set S of vectors (not necessarily finite) is defined to be the intersection W of all subspaces of V that contain S. W is referred to as the subspace spanned by S, or by the vectors in S. Conversely, S is called a spanning set of W, and we say that S spans W.
Alternatively, the span of S may be defined as the set of all finite linear combinations of elements of S, which follows from the above definition.
In particular, if S is a finite subset of V, then the span of S is the set of all linear combinations of the elements of S. In the case of infinite S, infinite linear combinations (i.e. where a combination may involve an infinite sum, assuming such sums are defined somehow, e.g. if V is a Banach space) are excluded by the definition; a generalization that allows these is not equivalent.
The real vector space R3 has {(-1,0,0), (0,1,0), (0,0,1)} as a spanning set. This particular spanning set is also a basis. If (-1,0,0) were replaced by (1,0,0), it would also form the canonical basis of R3.
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