Under what condition is a set {a1, ..., an} called a spanning set of V?

A set of vectors {a1, ..., an} is called a spanning set of a vector space V if every vector b in V can be expressed as a linear combination of the vectors in that set. This means that for any vector b within the space, there exist scalars c1, c2, ..., cn such that b = c1a1 + c2a2 + ... + cn*an. This property is fundamental in linear algebra because it defines the relationship between the set of vectors and the entire space they are meant to cover.

When a set spans a vector space, it ensures that you can reach any point (vector) in that space through a combination of the vectors in the set. This idea is crucial for understanding how vector spaces are structured and how bases and dimensions are defined within them. Other conditions, such as orthogonality or linear independence, do not necessarily apply to the concept of spanning a vector space. Hence, the condition identified accurately captures the essence of what it means for a set to span a vector space.