A spanning set in linear algebra is a collection of vectors that can generate every vector in a given vector space by forming linear combinations of those vectors. Put another way, any vector in the space can be expressed as a sum of scalar multiples of the vectors in the set. The notion of a spanning set is central to many topics, including the study of subspaces, linear independence, and basis construction.
Definition
A spanning set for a vector space (V) over a field (F) is a subset (S \subseteq V) such that every element of (V) can be written as a linear combination of elements of (S). Formally, if (S = {\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k}), then for any (\mathbf{w} \in V) there exist scalars (c_1, c_2, \dots, c_k \in F) satisfying
[ \mathbf{w} = c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k. ]
The set (
