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Let $W$ be a vector subspace of $F_2^n$. Then, $|W| = 2^k$ for $k \leq n$. Is it always true that $\text{dim}(W) = k$? If it is, where can I find a proof?

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    It may be easier to go the other way. If $W$ has a basis consisting $k$ vectors, how many elements does it have? Try it this way, if you are stuck!2011-08-31

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