Fill in the blank: A span of a set of vectors is the ____ subspace of V containing the vectors in the set.

The term that correctly fills in the blank is "smallest." The span of a set of vectors refers to the smallest subspace of the vector space ( V ) that contains all the vectors in that set. This means that it includes all possible linear combinations of those vectors and is the minimal set of vectors necessary to create the span.

By definition, a subspace must be closed under addition and scalar multiplication. The span fulfills this requirement, ensuring that any combination of the vectors in the set is included within this subspace. Thus, it serves as the smallest possible subset of ( V ) that can include the original vectors while preserving the properties of a vector space.

The option indicating "largest" would imply that numerous extra vectors or dimensions are included beyond those necessary to encompass the original set, which is not accurate in this context. "Unique" does not apply, as there can be many different spans depending on the initial set of vectors chosen, and "basic" does not convey the fundamental property of the span being minimal. Therefore, "smallest" is the most accurate description of the span of a set of vectors in linear algebra.