# Problems with the linear span of the empty set

I know that the span of the empty list $()$ or the empty $Ø$ set is defined to be the 'zero vector', what i don't get, is, the zero vector from what space? because all vector spaces contain the empty set, so $span() = {0} = {(0,0)} = {(0,0,0)} = ...$ and so on, but clearly $0 ≠ (0,0) ≠ (0,0,0)$ and so on. I'm really stuck on this. Hope someone understands my question. Thanks in advcance.

• What is the span of a set without referencing a vector space? – user223391 Jan 16 '17 at 2:18
• This question has a good explanation. – Fabio Somenzi Jan 16 '17 at 2:18

When you say "span", you have to specify the vector space $V$ you are talking about. To be completely formal, you would write $\operatorname{span}_V$. Often the $V$ is understood from context and we just write $\operatorname{span}$, but you should remember that it's always there.
"Span" only makes sense with respect to a given vector space. $\operatorname{span}\{ \emptyset\}$ is defined with respect to an underlying space, so whatever the zero vector is in the given space.