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How do I test for linear independence if there are only 3 equations in R4?

  1. Mar 3, 2012 #1
    1. The problem statement, all variables and given/known data

    So the dimension is R4. V1=[3 1 1 2], V2=[-2 -1 2 2] and V3=[2 1 2 1]

    2. Relevant equations

    3. The attempt at a solution

    The only way I know of to test for convergence is to make a matrix out of the row vectors of the vectors above (with the row vectors becoming the respective columns of the matrix) and to take the determinant. If det=0 then it is linearly dependent.

    The problem here, is that you get the following matrix: [3 -2 2; 1 -1 1; 1 2 2; 2 2 1] which, since it is 4x3, is not possible to find the determinant.

    Are there other ways to test for dependence?
  2. jcsd
  3. Mar 3, 2012 #2


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    Staff Emeritus
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    You could try using the definition of linear dependence. It will give you four equations and three variables... solve for the variables using the first three equations and see if you can satisfy the fourth still
  4. Mar 3, 2012 #3
    Think about trying to solve aV1+bV2+cV3=0

    Basically the same idea, what do a, b, and c have to be for V1, V2, and V3 to be linearly independent?
  5. Mar 3, 2012 #4
    Oh, I see. So then I write out the equations, put them into a matrix. I get the following.

    M=[3 -2 2 0; 1 -1 1 0; 1 2 2 0; 2 2 1 0].

    Brought to rref I get:

    M=[1 0 0 0; 0 1 0 0; 0 0 1 0; 0 0 0 0]. So then, since there is only a trivial solution, that means that they are all linear independent right? Just trying to make sure I understand.
  6. Mar 3, 2012 #5
    assuming you did it correctly, than that shows that the vectors are linearly independent
  7. Mar 4, 2012 #6
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