Question

Define Linear Span of vectors​

Answer 1

n linear algebra, the linear span of a set of vectors in a vector space is the intersection of all linear subspaces which each contain every vector in that set. The linear span of a set of vectors is therefore a vector space. Spans can be generalized to matroids and modules.

Answer 2

• 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.

Hope it will be helpful.