Proving that W1 \cap W2 is a Subspace of V

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SUMMARY

W1 ∩ W2 is a subspace of the vector space V if W1 and W2 are both subspaces of V. To prove this, one must verify three conditions: the zero vector is in W, W is closed under vector addition, and W is closed under scalar multiplication. The intersection W1 ∩ W2 consists of all vectors that belong to both W1 and W2, thereby satisfying the subspace criteria if both W1 and W2 meet the necessary conditions.

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Prove that if W1 and W2 are subspaces of the vector space V, then W1 \cap W2 is also a subspace of V.

Attempt at solution:
I really don't even know where to start on this because I am confused about how to prove in general that something is a subspace. Also, I don't know how to find what W1 \cap W2 is. Please help!
 
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mandygirl22 said:
Prove that if W1 and W2 are subspaces of the vector space V, then W1 \cap W2 is also a subspace of V.

Attempt at solution:
I really don't even know where to start on this because I am confused about how to prove in general that something is a subspace. Also, I don't know how to find what W1 \cap W2 is. Please help!

W1 \cap W2 consists of all the vectors that are in W1 and in W2.

To make it simpler to write, let's W = W1 \cap W2.

What are the things you need to check to verify that a set is a subspace of a given vector space?
 

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