In terms of spanning sets, when is a set considered minimal?

A set is considered minimal if it meets the criterion of having the least number of vectors necessary to span a particular vector space while still being able to generate the entire space. This concept is tied to the idea of linear dependence and independence. A spanning set can include redundant vectors—vectors that do not contribute to the span beyond what is already covered by other vectors in the set.

When a set is minimal, it means that if any vector is removed from the set, the remaining vectors would no longer be able to span the same space. Therefore, it effectively minimizes the cardinality or the number of vectors involved in spanning the same space without losing the ability to represent all points within that space.

In contrast, the other options do not align with this definition of minimality. Including only one non-zero vector does not necessarily span all spaces, and having no additional vectors is not a sufficient criterion; minimality is defined in relation to the total number of vectors. Lastly, maximizing dimension relates to spanning larger spaces but does not pertain to the minimality of the set. Thus, the correct understanding emphasizes the least number of vectors required, which is the essence of minimal spanning sets.