So I've seen two definitions of this:
Let $V$ be a vector space with subspace $W$. We say that $X \subseteq V$ spans $W$ if and only if
(Definition 1): Every $\vec{w} \in W$ can be written as a linear combination of vectors in $X$.
(Definition 2): $span(X) = W$.
I don't think the two definitions are equivalent are they? Clearly the condition in Def. 2 implies the condition in Def. 1, but not the other way round.
For example take $V$ to be the vector space of ordered pairs over $\mathbb{Z}_2$, then if $W = \{(0,0),(1,1)\}$ and $X=\{(0,1),(1,0)\}$, $X$ spans $W$ according to the first definition, but not the second.
What then would you have understood if I were to tell you that "$X$ spans $W$"? Would you have agreed or disagreed with me?
I know the usual drill is "clarify your definitions from the start" in whatever work you're doing, and that as long as you do that it will be fine, but I'm just wanting to know what peoples' immediate interpretation would be. Because it does impact a little on the reading and understanding process.
(For the record I'm a fan of Definition 1.)