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I Decomposition Of A Vector

  1. Jul 22, 2016 #1
    Hello,

    I am trying to figure out how to best decompose a vector into a best fit linear superposition of other, given vectors.

    For instance is there a way of finding the best linear sum of:

    (3,5,7,0,1)
    (0,0,4,5,7)
    (8,9,2,0,4)

    That most closely gives you (1,2,3,4,5)

    My problem contains more, higher order vectors so if there is a general statistical way of doing a decomposition like this that would be great.

    Thanks!
     
  2. jcsd
  3. Jul 22, 2016 #2

    blue_leaf77

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    You can use least square solution. First, realize that you can express a linear combination of ##n## ##m\times 1## column vectors as a matrix product between a matrix formed by placing those ##n## columns next to each other and a ##n \times 1## column vector consisting of the coefficients of each vector in the sum. Denote the first matrix as ##A## and the second (column) one as ##x##, you are to find ##x## such that ##||Ax-b||## is minimized where ##b## is the ##m \times 1## column vector you want to fit to.
     
  4. Jul 22, 2016 #3

    BvU

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    My hunch was that the three vectors span a 3D space in which you can express the part of (1,2,3,4,5) that lies in that space exactly (by projections). For the two other dimensions there's nothing you can do. Am I deceiving myself ?
     
  5. Jul 25, 2016 #4

    chiro

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    Hey dman12.

    This is equivalent to solving the linear system in RREF.

    Understanding this process of row reduction and why it works will help you understand a lot of linear algebra in a practical capacity.
     
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