What can be assumed if a set of vectors spans a space?

When a set of vectors spans a space, it means that every vector in that space can be expressed as a linear combination of the vectors in the set. This is the defining property of a spanning set, as it guarantees that any vector within the space can be constructed using the available vectors by appropriately scaling and adding them together.

In the context of linear algebra, if you have a vector space, a spanning set forms the foundation of that space, ensuring that the entirety of that space can be generated from the linear combinations of those vectors. Therefore, the correct assertion about spanning is that it allows for the construction of any vector in the space from the set itself.

The other statements provide conditions that do not necessarily follow from the definition of spanning. For example, saying that all vectors are linearly dependent doesn’t hold true for spanning sets, as they can include independent vectors. Furthermore, having a dimension exceeded or an infinite number of vectors does not inherently relate to the ability to span a space either, as a finite spanning set can still adequately cover the space as long as its cardinality matches the dimension of that space.