Theorems
Theorem 1: The subspace spanned by a non-empty subset S of a vector space V is the set of all linear combinations of vectors in S.
This theorem is so well known that at times it is referred to as the definition of span of a set.
Theorem 2: Every spanning set S of a vector space V must contain at least as many elements as any linearly independent set of vectors from V.
Theorem 3: Let V be a finite dimensional vector space. Any set of vectors that spans V can be reduced to a basis for V by discarding vectors if necessary (i.e. if there are linearly dependent vectors in the set). If the axiom of choice holds, this is true without the assumption that V has finite dimension.
This also indicates that a basis is a minimal spanning set when V is finite dimensional.
Read more about this topic: Linear Span