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Intersection of subspaces

  1. Sep 8, 2005 #1
    Hi can someone please help me with the following question. Such questions always trouble me because I don't know where to start and/or cannot continue after starting.

    Q. Let H and K be subspaces of a vector space V. Prove that the intersection of K and H is a subspace of V.

    By the way that the question is set out I figure that all I need to show is that the intersection of K and H is non-empty, closed under scalar multiplication and addition. So here is what I've tried.

    H and K are subspaces of the vector space V so they both contain the zero vector. So it follows that the intersection contains the zeor vector so that [tex]H \cap K \ne \emptyset [/tex].

    That's all I can think of. I'm not sure if I can make any other assumptions about vectors which are common to H and K and so I cannot continue.
  2. jcsd
  3. Sep 8, 2005 #2


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    Now show that a lineair combination of vectors of [tex]H \cap K[/tex] is still in [tex]H \cap K[/tex].
  4. Sep 8, 2005 #3
    Thanks for your response but that's the sort of thing that I'm having trouble with. All I've been able to show is that the zero vector is in the intersection. I don't know which are vectors are in the intersection. I cannot figure out what else I extract from the stem of the question to assist me. It's probably just a conceptual thing but I can't really see what I can and can't use.
  5. Sep 8, 2005 #4


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    Well you already showed it's not empty. Now take the scalars [itex]\alpha ,\beta \in \mathbb{R}[/itex] (or any other field of course) and the vectors [itex]\vec x,\vec y \in H \cap K[/tex]. Now, since the vectors are in both subspaces, we can say that:
    [tex]\alpha \vec x + \beta \vec y \in H[/tex]
    [tex]\alpha \vec x + \beta \vec y \in K[/tex]

    And thus: [tex]\alpha \vec x + \beta \vec y \in H \cap K[/tex]
  6. Sep 8, 2005 #5
    Ok thanks for the help.
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