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Homework Help: Find a Basis for the solution set

  1. Sep 26, 2010 #1
    3x1 + x2 + x3 = 0
    6x1 + 2x2 + 2x3 = 0
    -9x1 - 3x2 - 3x3 = 0

    I'm not sure how to approach this problem. I've rewritten these equations as a matrix

    [3 1 1]
    [6 2 2]
    [-9 -3 -3]

    Reduced Echelon from gave me this
    [3 1 1]
    [0 0 0]
    [0 0 0]

    Am I approaching this the wrong way?
    What should I do next? Please help.
    Thank You.
    Last edited: Sep 26, 2010
  2. jcsd
  3. Sep 26, 2010 #2


    Staff: Mentor

    By inspection it can be seen that the 2nd equation is 2 times the first, and the 3rd is -3 times the first. In essence, you have the same equation written three times.
    Solve the first equation to get
    x1 = -(1/3)x2 - (1/3)x3
    x2 = x2 + 0x3
    x3 = 0x2 + x3

    The 2nd and 3rd equations above are obviously true.

    If you stare at this system awhile, you might see that any vector <x1, x2, x3> in this set can be written as a linear combination of two vectors that happen to be linearly independent.
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