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If it must, why? If it mustn't, why? I was able to prove that a certain set of vectors doesn't have the zero vector as a member and normally, I've been told to conclude that it is not a vector subspace from this point. I got confused and wondered if this always shows that the set isn't an empty set since that is one of the conditions a vector subspace must satisfy.

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Have you looked at the definition of a subspace lately? It must be a vector space in its own right, which necessitates a zero vector, which must coincide with the zero vector of the original space. – anon Oct 16 '12 at 19:25
up vote 7 down vote accepted

Let $V$ be a vector space and $U\subset V$ be a subspace. Since $U$ is non-empty there is a $u\in U$. Since $U$ is a subspace we get $-u\in U$. Hence also $0=u+(-u)\in U$.

Alternatively also $0=0_K\cdot u\in U$, where $0_K$ denotes the zero in the ground field $K$ (assuming $V$ is a $K$-vector space).

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a simple and straight forward proof! – user31280 Oct 16 '12 at 19:37

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