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Q. Let H and K be subspaces of a vector space V. Prove that the intersection of K and H is a subspace of V.

By the way that the question is set out I figure that all I need to show is that the intersection of K and H is non-empty, closed under scalar multiplication and addition. So here is what I've tried.

H and K are subspaces of the vector space V so they both contain the zero vector. So it follows that the intersection contains the zeor vector so that [tex]H \cap K \ne \emptyset [/tex].

That's all I can think of. I'm not sure if I can make any other assumptions about vectors which are common to H and K and so I cannot continue.