MATH 110 - SOLUTION TO QUIZ 1 LECTURE 1, SUMMER 2009 GSI: SANTIAGO CA ˜ NEZ Let V be a vector space over a ﬁeld F , and let U and W be subspaces of V . Prove that U ∩ W is a subspace of V . Proof. First, since U and W are both subspaces of V , we know that 0 ∈ U and 0 ∈ W . Hence 0 ∈ U ∩ W . Now, let x,y ∈ U ∩ W . Then x,y ∈ U and x,y ∈ W . Since U is closed under addition, we have that x + y ∈ U , and similarly since W is closed under addition, x + y ∈ W . Thus x + y ∈ U ∩ W so U ∩ W is closed under addition.
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This note was uploaded on 08/31/2009 for the course MATH 110 taught by Professor Gurevitch during the Summer '08 term at University of California, Berkeley.