# Linear span of the empty set

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?

• This also implies $\rm Span \{ ~ \mathbf 0~ \} = Span \{ \}$
– john
Mar 28, 2018 at 5:50

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.