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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.

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

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"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.

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