C3093218 - V Since both U and V are subspaces x y ∈ U and...

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5. (Statement of the problem.) Proof. Since 0 U and 0 V , 0 U V . So U V is nonempty. Take x U V and a scalar α . Then x U and x V . Since both U and V are subspaces, α x U and α x V . So α x U V . U V is closed under scalar mutiplication. Take x , y U V . Then x , y U and x , y
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Unformatted text preview: V . Since both U and V are subspaces, x + y ∈ U and x + y ∈ V . So x + y ∈ U ∩ V . U ∩ V is closed under addition. By the definition, U ∩ V is a subspace of W . 1...
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