Prove that any subspace, W, of a finite-dimensional vector space V must also be finite dimensional.
Prove subspace of finite dimensional vector space is finite dimensional
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$\begingroup$
linear-algebra
proof-writing
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0You don't have to use the $p$lus-minus theorem, but if you really want to, try to think of a set of vectors to apply the "plus" or the "minus" to yield a contradiction with *W* being finite dimensional. – 2012-11-19
2 Answers
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Consider a basis $\beta$ of $W$, then $\beta$ is linearly independent in $V$ so you can extend $\beta$ to $\gamma$ a basis of $V$, so $\gamma$ must be finite, by hypothesis.
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HINT: Let $n=\dim\ V$. What is the maximum size of any linearly independent subset of $V$?