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Thread: 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 $\displaystyle V_1 $ and $\displaystyle V_2$ be subspaces of $\displaystyle \Omega$ and let $\displaystyle V_0 = V_1 \cap V_2$ Under what conditions $\displaystyle P_{V_0} =P_{V_1} P_{V_2}$


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

    2. Let $\displaystyle V_1,V_2,V_3$ be subspaces. Does $\displaystyle V_1 \cap (V_2 + V_3) = V_1\cap V_2 + V_1\cap V_3$ in general? If not, does it hold if $\displaystyle V_2$ and $\displaystyle V_3$ 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 $\displaystyle V_1 $ and $\displaystyle V_2$ be subspaces of $\displaystyle \Omega$ and let $\displaystyle V_0 = V_1 \cap V_2$ Under what conditions $\displaystyle P_{V_0} =P_{V_1} P_{V_2}$


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

    2. Let $\displaystyle V_1,V_2,V_3$ be subspaces. Does $\displaystyle V_1 \cap (V_2 + V_3) = V_1\cap V_2 + V_1\cap V_3$ in general? If not, does it hold if $\displaystyle V_2$ and $\displaystyle V_3$ 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 $\displaystyle V_2+V_3=V$ (where $\displaystyle V$ is the full space) so that the LHS is $\displaystyle V$ but $\displaystyle V_1\cap V_2=V_1\cap V_3=\varnothing$? Can that happen?
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