Span of an empty list Its a simple question . But still I thought of asking it coz I don't see the logic behind it. Recently I read a statement "the Span of empty list ( ) equals {0}" .What we know is an empty list has no elements in it,so how can we even talk of a span of it as span is a linear combination of all the vectors in a list and in empty list no vector is present at all. So, is it kind of just a declaration that we make just like that?? i guess there is no logic behind it and we just assume it like that. But why? What's the reason of doing so? And why do we need to declare a span of an empty list anyway?? Sorry I am not getting the logic behind it and that's why I thought of asking you guys. Thanks.
 A: The singleton $V=\{0\}$ satisfies all of the axioms of a vector space over any field, when addition and scalar multiplication are defined in the only way they can be.  What are the natural answers to the following questions?


*

*Is $V$ finite-dimensional?  If so, what is its dimension?


Since $V$ is a subspace of all vector spaces (including one-dimensional vector spaces), it makes sense to answer that $V$ is indeed finite-dimensional and that its dimension is $0$.  Next, if the dimension of a vector space is equal to the cardinality of a basis, we are only left with the option of declaring the set $\varnothing$ to be a basis for $V$.  Since a basis is a spanning set, it follows that we have $\operatorname{Span}\varnothing=\{0\}$.
A: You can just assume it (or more correctly define it) that way.
That being said the span of a set $S \subset V$ in a vector space $V$ is just the smallest subspace of $V$ containing $S$, as such $\{0\}$ is the span of $\emptyset$. (Note that any subspace must contain $0$ by definition).
In terms of linear combinations you could think of the span of a set $S \subset V$ as the set $\{\lambda_1v_1 + ...\lambda_nv_n + 0 : n \in \mathbb N , v_1,...,v_n \in S, \lambda_1,...,\lambda_n \in \mathbb R\}$ (assuming $\mathbb R$ is the underlying field). This definition accommodates the possibility that the set $S$ is empty, as well as the possibility that the set $S$ is infinite.
A: It makes sense if you think about the span of a set of vectors as the smallest vector space containing the set of vectors. What is the smallest vector space containing the empty set? It's the smallest vector space since all vector spaces contain the empty set. Therefore, the span of the empty set is the zero vector space.
