According to the definition of a subset, when is 'S' a subspace of 'V'?

A set 'S' is considered a subspace of a vector space 'V' if it satisfies certain criteria. One of the fundamental requirements is that it must be closed under vector addition (often referred to as "o-plus") and scalar multiplication ("o-dot"). This means that if you take any two vectors from 'S' and add them together, the resulting vector must also be in 'S'. Similarly, if you take any vector from 'S' and multiply it by a scalar, the result must still belong to 'S'.

This property ensures that 'S' retains the structure of a vector space within 'V'. If this closure property is satisfied, along with containing the zero vector and being non-empty, 'S' qualifies as a subspace.

While the other options address different aspects of sets and vector spaces, they do not align with the necessary criteria for a subspace. For instance, containing all vectors of 'V' is unnecessary and overly restrictive, as a subspace could be any smaller vector space, not just the entire space itself. Simply including at least one vector does not ensure that 'S' behaves like a vector space, as more structure is needed. Lastly, the concept of exceeding the size of 'V