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Linear Dependence/Independence

  1. Apr 21, 2010 #1
    1. The problem statement, all variables and given/known data
    Is the set S = {(3,-2,4),(4,6,-4),(3,6,-2),(-13,2,-18)} linearly dependent? If so, give a dependency equation.


    2. Relevant equations



    3. The attempt at a solution

    I first placed the set S into a matrix equation
    [3 4 3 -13
    -2 6 6 2
    4 -4 -2 -18]

    then put it into rref:

    [1 0 0 5.44 0
    0 1 0 7.76 0
    0 0 1 -9.24 0]

    Since the final column is all zeros, is the system linearly independent?
     
  2. jcsd
  3. Apr 21, 2010 #2

    radou

    User Avatar
    Homework Helper

    Can a set of more than 3 vectors in R^3 be linearly independent?
     
  4. Apr 21, 2010 #3
    If the row rank is less than the number of variables then there has to be a parametric solution. How would I write a dependency equation, then?
     
  5. Apr 21, 2010 #4

    Mark44

    Staff: Mentor

    This matrix,
    [3 4 3 -13]
    [-2 6 6 2]
    [4 -4 -2 -18]
    represents the vector equation c1*v1 + c2*v2 + c3*v3 + c4*v4 = 0. As a matrix equation this is Ac = 0, where the columns of A are your four vectors, and c = <c1, c2, c3, c4>^T.

    Assuming that your work is correct and that you ended with the next matrix (I removed the 5th column of 0s),

    [1 0 0 5.44]
    [0 1 0 7.76]
    [0 0 1 -9.24]

    this matrix says that c1 + 5.44*c4 = 0, c2 + 7.76*c4 = 0, and c3 - 9.24*c4 = 0.
     
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