[Groxx]

Yeah, it took a while to sink into my head that many of these "wait, why is span({}) = {0}?" kinds of cases have answers that sum up as "because anything else means other rules are inconsistent, and the whole thing is either less useful or useless". It's "arbitrary", but it's either the only useful option, or sometimes a simple(st) one of many.

Even just one number theory course helped a lot, since it brought that kind of consistency into its own concept, where [this set of rules] forms a ring, and [this set] forms a field, etc.

[Koshkin]

To be fair, in this case the rule is not quite arbitrary, because it, in fact, follows from the definition of span (as a subspace).