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Does it make any sense to talk about the dimension of a subset (not necessarily a subspace) of a vector space at all? What if say you can pick $n$ linearly independent vector from the subset but the set is not closed, then how do we call this number $n$? (Sorry, I know that my question maybe poorly worded, I hope you understand my meaning.) Thanks.

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It sounds like what you're after is the dimension of the span of the subset. Given a vector space $V$ and some subset $S\subseteq V$, not necessarily a subspace of $V$, then $\operatorname{span}(S)$ is the smallest subspace of $V$ that contains $S$. Thus, for example, if one can find $n$ linearly independent vectors in $S$, then we have also found $n$ linearly independent vectors in $\operatorname{span}(S)$ (because $S\subseteq\operatorname{span}(S)$), hence $$\dim(\operatorname{span}(S))\geq n.$$ If $n$ is the largest such number, then in fact we have equality.

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Thank you, Zev! – glinka Nov 19 '11 at 23:42

It certainly makes sense to speak of the dimension of the subspace spanned by your subset, but you should probably say so explicitly, because there are other possible meanings of "dimension" that could otherwise be confused.

For example, the subset $$A = \{(x,y,z)\mid x=0\land y^2+z^2=1\} \subseteq \mathbb R^3$$ spans a plane (of dimension 2), but $A$ is also a manifold of dimension $1$, and if you just speak of the "dimension of $A$", it is likely that people will understand the latter rather than the former.

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Thanks, Henning! – glinka Nov 19 '11 at 23:42

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