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Find linear dependence on these vectors

  1. Oct 23, 2012 #1
    1. The problem statement, all variables and given/known data
    Suppose V = R^4 and let U = <X>, where X = {(1,0,-2,1),(2,-2,0,3),(0,2,-4,-1),(-1,2,-2,-2)}
    Find linear dependence on X and use it to find a smaller generating set of U. Repeat the step until you reach a basis for U.


    2. Relevant equations



    3. The attempt at a solution

    I have formed a matrix of the 4 vectors in X and reduced it to echelon form. I got
    (1 | 0 | 2 | 1
    0 | 1 | -1 | -1
    0 | 0 | 0 | 0
    0 | 0 | 0 | 0)

    Let's say the 4 vectors were s, t, u, v respectively.
    Then xs + xt + yu + zv = 0 is true for some constants w,x,y,z (1)

    Then according to the REF form we can form two equations: w = 2y +z and x = -y+z
    I thought I could substitute these into equation (1) and end up showing one vector as a combination of the others but I am not able to reach anywhere. Where am I wrong? Is there a better method to go about this?
     
  2. jcsd
  3. Oct 23, 2012 #2

    haruspex

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    I would have thought that the relationship was implied by the steps to get the echelon form. I.e. in producing the null rows, you effectively executed the equation you're looking for.
     
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