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

[buckaroobill]

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

Let \(\displaystyle V\) be a vector space and let \(\displaystyle W_1\) and \(\displaystyle W_2\)be subspaces of \(\displaystyle V\). Prove that \(\displaystyle W_1 \cap W_2\) is a subspace of \(\displaystyle V\). (Do not forget to show that \(\displaystyle W_1 \cap W_2\) is nonempty.)

 

[daon]

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.