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Possible Duplicate:
Linear Algebra: If A spans B, does B necessarily span A if dim A = dim B?

Thanks for looking at my question!

Given:

I: $\{b_1,...,b_n\}$ linear independent vectors and $\{c_1,...,c_n\}$ linear independent vectors.

II: $\{b_1,...,b_n\}$ span subspace spanned by $\{c_1,...,c_n\}$

Does $\{c_1,...,c_n\}$ necessarily span the subspace spanned by $\{b_1,...,b_n\}$?

I'm pretty sure it does, but am not sure how to prove it. Also, is there any better way to write this? It's certainly less than poetic.

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    Please do not cross-post without indicating that you did; this migrated duplicate is but one example of the inefficiencies this can cause, the most obvious one being that people will work separately in different threads without seeing each other's results and will thus needlessly duplicate each other's efforts.2012-12-12
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    Are you seriously asking whether $$ = $$ $\Rightarrow$ $$ = $$?2012-12-12
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    Or do you mean whether $\subset \Rightarrow \subset $?2012-12-12

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