Linear span of the empty set

If you want to stay coherent, you almost never have a choice for such "empty" definitions.

Here, the span of $X$ is the set of linear combinations $\sum_{x\in X} \lambda_x x$. So the question boils down to what is an empty sum. It has to be $0$, because when you add an empty sum to $s$, you want to get $s$. An empty operation is always the neutral element for this operation, like an empty product is $1$.

So here, $Span(\emptyset)$ is the set of all possible empty sums, which is $\{0\}$. It is also a good remark that this is coherent with the fact that for any set $S$, $Span(S)$ is a vector space.

The linear span of a subset of a vector space is the smallest linear subspace which contains it or, equivalently, the intersection of all the linear subspaces which contain it.

If you carry out this for the empty set, you'll get the correct result.