Linear algebra proof: Let V be a vector space, and let....

buckaroobill

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The following proof was confusing me so any help would be appreciated!

Let V\displaystyle V be a vector space and let W1\displaystyle W_1 and W2\displaystyle W_2be subspaces of V\displaystyle V. Prove that W1W2\displaystyle W_1 \cap W_2 is a subspace of V\displaystyle V. (Do not forget to show that W1W2\displaystyle W_1 \cap W_2 is nonempty.)
 
If a,b are in the intersection, then a,b are in W1 and a,b are in W2. Therefore a+b must also be in W1,W2, so it lies in the intersection. Let k be a scalar and v be a vector in the intersection of W1,W2. Then v is in W1 and v is in W2, and so kv is in W1, W2 and hence it also lies in the intersection. It is nonempty because the zero vector for V is in any subspace of V, so it must be in the intersection of any set of subspaces from V.
 
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