If the span of {a1, ..., an} equals V, what can we conclude about the set?

When the span of the set ({a_1, \ldots, a_n}) equals the vector space (V), this means that every vector in (V) can be expressed as a linear combination of the vectors in the set. Therefore, the set ({a_1, \ldots, a_n}) serves as a spanning set for (V). This indicates that the set contains enough vectors to cover the entire space (V).

Spanning sets are critical in linear algebra because they provide a way to represent every element of a given vector space through a combination of the vectors included in the set. Although this set might not necessarily be linearly independent or complete in the sense of forming a basis (which would also require linear independence), it does fulfill the essential characteristic of covering the entire space.

In contrast, a linearly independent set does not necessarily span the entire space; thus, stating that the set is linearly independent does not follow from the span condition. Additionally, the inclusion of the zero vector is not a requirement for spanning; a set can span a space without including zero. Finally, the conclusion that ({a_1, \ldots, a_n