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Decomposing a vector

  1. Nov 28, 2013 #1
    1. The problem statement, all variables and given/known data
    There is a subspace that contains all the vectors in the form (x, 2y, x). Decompose the vector (2, 3, -1) into a sum of an element from the orthogonal complement of this subspace and an element from the subspace. Find the distance from (2, 3, -1) to this subspace.

    3. The attempt at a solution
    To find the orthogonal complement of this subspace, I found the kernel, which in this case happens to only contain the zero vector. That means only a particular solution exists, but obviously (2, 3, -1) is not a particular solution, so I'm not sure how to decompose this, much less find the distance.
     
  2. jcsd
  3. Nov 28, 2013 #2

    LCKurtz

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    Why do you say the kernel is only the zero vector? ##(x,2y,x)=x(1,0,1)+y(0,2,0)## is clearly only two dimensional.
     
    Last edited: Nov 28, 2013
  4. Nov 28, 2013 #3

    vela

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    Found the kernel of what?
     
  5. Nov 28, 2013 #4
    The kernel of
    ##\ \left( \begin{array}{ccc}
    1 & 0 \\
    0 & 1 \\
    1 & 0 \end{array} \right)##
     
  6. Nov 28, 2013 #5

    vela

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    Your domain should be R3, not R2.
     
  7. Nov 28, 2013 #6
    So then it's:
    ##\ \left( \begin{array}{ccc}
    1 & 0 & 0 \\
    0 & 1 & 0 \\
    1 & 0 & 0 \end{array} \right)##
    ?
     
  8. Nov 28, 2013 #7

    LCKurtz

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    Yes, that matrix represents the transformation ##(x,y,z)\rightarrow (x,2y,x)##.

    [Edit]: There would be a 2 in the center.
     
    Last edited: Nov 28, 2013
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