By definition, the span of the empty set is the zero vector.
(kind of makes sense if we understand the concept of the empty sum
evaluating to zero). Thus, your basis is: the empty set.
I.e. it is a singleton.
I've already showed that it is a subspace of (hopefully you'll agree that it is indeed a vector space).
But the problem is:
How can I find a basis for it?!
A basis must span the set S, but it must also be a linearly independent set!
So if I understand correctly, spans S, but it is not linearly independent (since any singleton set containing just the 0 vector is linearly dependent).
What am I to do?