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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.