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Math Help - Proving intersection is a subspace

  1. #1
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    Proving intersection is a subspace

    Let W_1 and W_2 be two subspaces of a vector space V over the field \mathbb{F}. Prove that the intersection of W_1 and W_2 is a subspace of V.


    Can someone check my working for this question

    Let x\in W_1, y\in W_1

    Since W_1 is a subspace, it is closed under addition and so x+y\in W_1

    Similarly, let x\in W_2, y\in W_2

    Since W_2 is a subspace, it is closed under addition and so x+y\in W_2

    \Rightarrow x+y\in W_1\cap W_2\Rightarrow Closure under addition


    Let \lambda\in\mathbb{R}, x\in W_1, W_2

    \lambda x\in W_1 since W_1 is a subspace

    \lambda x\in W_2 since W_2 is a subspace

    Clearly, \lambda x\in W_1\cap W_2\Rightarrow Closure under scalar multiplication

    \Therefore W_1 \cap W_2 is a subspace of V




    What about the condition for non-empty? This condition always confuses me. Why is it that the zero vector has to be included for it to be non-empty?
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by acevipa View Post
    Let W_1 and W_2 be two subspaces of a vector space V over the field \mathbb{F}. Prove that the intersection of W_1 and W_2 is a subspace of V.


    Can someone check my working for this question

    Let x\in W_1, y\in W_1

    Since W_1 is a subspace, it is closed under addition and so x+y\in W_1

    Similarly, let x\in W_2, y\in W_2

    Since W_2 is a subspace, it is closed under addition and so x+y\in W_2

    \Rightarrow x+y\in W_1\cap W_2\Rightarrow Closure under addition


    Let \lambda\in\mathbb{R}, x\in W_1, W_2

    \lambda x\in W_1 since W_1 is a subspace

    \lambda x\in W_2 since W_2 is a subspace

    Clearly, \lambda x\in W_1\cap W_2\Rightarrow Closure under scalar multiplication

    \Therefore W_1 \cap W_2 is a subspace of V




    What about the condition for non-empty? This condition always confuses me. Why is it that the zero vector has to be included for it to be non-empty?
    well, since \displaystyle W_1 and \displaystyle W_2 are subspaces, they each contain 0. so 0 will be in their intersection, and hence, the intersection is non-empty.

    containing zero is important for vector spaces and are hence important for subspaces. so the subspaces must have 0 in them. it is not to say that they have to have 0 to be non-empty per sae, but it is usually easy to verify that the zero vector is in there. so you always check that. and if 0 is not in there, then you're in trouble. you wouldn't have a vector space at all. just some other kind of set.
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  3. #3
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    Thanks a lot. I understand that well now.
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