What is necessary for a set of vectors to span a vector space?

For a set of vectors to span a vector space, it is essential that they cover all possible linear combinations of the space. This means that any vector within that vector space can be expressed as a linear combination of the spanning set. Essentially, the spanning set must be able to generate every vector in the space through linear combinations of the vectors it contains.

Spanning is a fundamental concept in linear algebra because it allows us to construct the entire vector space from a collection of vectors. If the vectors do not span the space, there will be vectors that cannot be created from the linear combinations of the given set, indicating that the set is insufficient to represent the entire space.

While linear independence, cardinality, and uniqueness of vectors are important properties in certain contexts, they do not directly ensure that the vectors span the space. A set can be linearly independent and still not span the entire space if it does not encompass enough vectors. Moreover, it's possible for a spanning set to contain linearly dependent vectors, as long as it is still capable of generating all vectors in the space. Hence, to determine if a set spans a vector space, one must focus on its ability to cover all linear combinations.