Linear span

In linear algebra, the linear span (also called the linear hull or just span) of a set S of vectors in a vector space is the smallest linear subspace that contain the set. It can be characterized either as the intersection of all linear subspaces that contain S, or as the set of linear combinations of elements of S. The linear span of a set of vectors is therefore a vector space. Spans can be generalized to matroids and modules. In linear algebra, the linear span (also called the linear hull or just span) of a set S of vectors in a vector space is the smallest linear subspace that contain the set. It can be characterized either as the intersection of all linear subspaces that contain S, or as the set of linear combinations of elements of S. The linear span of a set of vectors is therefore a vector space. Spans can be generalized to matroids and modules. For expressing that a vector space V is a span of a set S, one commonly uses the following phrases: S spans V; S generates V; V is spanned by S; V is generated by S; S is a spanning set of V; S is a generating set of V. Given a vector space V over a field K, the span of a set S of vectors (not necessarily infinite) 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 (vectors) 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. Another spanning set for the same space is given by {(1,2,3), (0,1,2), (−1,1/2,3), (1,1,1)}, but this set is not a basis, because it is linearly dependent. The set {(1,0,0), (0,1,0), (1,1,0)} is not a spanning set of R3; instead its span is the space of all vectors in R3 whose last component is zero. That space (the space of all vectors in R3 whose last component is zero) is also spanned by the set {(1,0,0), (0,1,0)}, as (1,1,0) is a linear combination of (1,0,0) and (0,1,0). It does, however, span R2. The empty set is a spanning set of {(0, 0, 0)} since the empty set is a subset of all possible vector spaces in R3, and {(0, 0, 0)} is the intersection of all of these vector spaces.