Linear span of the empty set

Question

Why is the span of the empty set defined to be $\{0\}$? It is known that the span of any nonempty set of vectors in a vector space $V$, gives a subspace of $V$, and it is stated in “Linear Algebra Done Right” by Axler, that to be consistent with this, the span of the empty set is defined to be $\{0\}$? Is this the only reason or does this definition prove useful in other ways later on?

Answer 1

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.

Answer 2

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.