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Let W be a subspace of the vector space V. u and v are elements of V such that u + v and u - v are in W. I need to prove that u and v are in W.

I think I can use this theorem, but I'm not quite sure if I can apply it in the reverse direction.

If W is a nonempty subset of a vector space V, then W is a subspace of V if and only if the following closure conditions hold. 1. If u and v are in W, then u + v is in W 2. If u is in W and c is any scalar, then cu is in W.

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  • $\begingroup$ "if and only if" means that you can use it in the reverse direction, namely if conditions 1 and 2 hold, then if W is nonempty subset of V, W is subspace of V. You don't need the reverse direction though, because the question already assumed that W is a subspace of V. $\endgroup$
    – jh4
    Nov 19, 2014 at 18:24

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Hint: $2u= (u+v)+(u-v)$, then use vector space property of $W$...

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