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Math Help - Vector subspaces and projection questions

  1. #1
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    Vector subspaces and projection questions

    I'm stuck with two questions on vector subspaces and projections.

    1. Let  V_1 and V_2 be subspaces of \Omega and let V_0 = V_1 \cap V_2 Under what conditions  P_{V_0} =P_{V_1} P_{V_2}


    Well, for this one I see that it will work if one of  V_1 and  V_2 contains the other and also if V_1 \perp V_2 , but this doesn't seem like a good answer.

    2. Let V_1,V_2,V_3 be subspaces. Does V_1 \cap (V_2 + V_3) = V_1\cap V_2 + V_1\cap V_3 in general? If not, does it hold if V_2 and V_3 are linearly independent?

    Thanks in advance,
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Re: Vector subspaces and projection questions

    Quote Originally Posted by akolman View Post
    I'm stuck with two questions on vector subspaces and projections.

    1. Let  V_1 and V_2 be subspaces of \Omega and let V_0 = V_1 \cap V_2 Under what conditions  P_{V_0} =P_{V_1} P_{V_2}


    Well, for this one I see that it will work if one of  V_1 and  V_2 contains the other and also if V_1 \perp V_2 , but this doesn't seem like a good answer.

    2. Let V_1,V_2,V_3 be subspaces. Does V_1 \cap (V_2 + V_3) = V_1\cap V_2 + V_1\cap V_3 in general? If not, does it hold if V_2 and V_3 are linearly independent?

    Thanks in advance,
    The answer to your first question can be found on my blog post here. Any ideas for the second? What if V_2+V_3=V (where V is the full space) so that the LHS is V but V_1\cap V_2=V_1\cap V_3=\varnothing? Can that happen?
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