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Math Help - [SOLVED] Projections question

  1. #1
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    [SOLVED] Projections question

    Can you obtain the same vector by first projecting y onto V, then projecting this vector on x as by projecting y directly on x or more generally , if V is a subspace and V_1 is a subspace of V, does p(p(\textbf{y}|V)|V_1) = p(\textbf{y}|V_1)

    From the way the question is phrased it seems like its true, but I was wondering if there was an intuitive explanation as to why.
    Last edited by jass10816; April 8th 2010 at 07:35 AM.
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  2. #2
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    Quote Originally Posted by jass10816 View Post
    Can you obtain the same vector by first projecting y onto V, then projecting this vector on x as by projecting y directly on x or more generally , if V is a subspace and V_1 is a subspace of V, does p(p(\textbf{y}|V)|V_1) = p(<b>y</b>|V_1)

    From the way the question is phrased it seems like its true, but I was wondering if there was an intuitive explanation as to why.
    I'm not sure about your question nor your notation. But let me guess: suppose we are given vectors x and y such that the direct projection of y on x gives a non-null vector. Now, I assume that, given a vector y like this, it is possible to find a subspace V such that y, when projected onto V, gives the null-vector. If so: we have rejected the above proposition, because no amount of further projecting that projection of y will ever give something other than the null-vector.
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  3. #3
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    Okay, but as long as the projection of y onto V is non-null, then the equality holds?
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  4. #4
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    Quote Originally Posted by jass10816 View Post
    Okay, but as long as the projection of y onto V is non-null, then the equality holds?
    Of course not. Consider a quite elementary case in \mathbb{R}^3: a vector y that has positive x_1, x_2 and x_3 components, say y=(1,1,1), gets first projected onto the x_2x_3-plane (your V), this reduces its x_1 component to 0. Then you project that projection (0,1,1) onto the x_1-axis (i.e. the direction of a vector that points in the positive direction of the x_1-axis): you get the null vector (0,0,0).
    But if you project that vector directly onto the x_1-axis, you get a non-null vector, namely (1,0,0).
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