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Proving Linear Independence of vectors

  1. Jan 26, 2014 #1
    1. The problem statement, all variables and given/known data

    Let x1 = (1, 2, -1, 1), x2 = (-1, -1, -1, -1), x3 = (1, 1, 1, 0), x4 = (-2, -1 -4 -1)

    Show that x1, x3 and x4 are linearly independent



    2. Relevant equations



    3. The attempt at a solution
    Now I used the equation:
    ax1+bx2+cx3+dx4=0

    Hence forth the augmented matrix of the equation is,

    [tex]
    \begin{pmatrix}
    1 & -1 & 1 & -2 & | & 0\\
    2 & -1 & 1 & -1 & | & 0\\
    -1 & -1 & 1 & -4 & | & 0\\
    1 & -1 & 0 & -1 & | & 0
    \end{pmatrix}
    [/tex]

    This is row reduced to,

    [tex]
    \begin{pmatrix}
    1 & -1 & 1 & -2 & | & 0\\
    0 & 1 & -1 & 3 & | & 0\\
    0 & 0 & -1 & 1 & | & 0\\
    0 & 0 & 0 & 0 & | & 0
    \end{pmatrix}
    [/tex]

    From as there is no leading entry corresponding to d, Setting d = t, the general solution is:

    a = -t, b = -2t, c = t, d = t.

    And as the number of leading entries =! number of unknowns so the vectors are linearly dependant

    This is the point where I get confused as the question asks how they are linearly independant so I am quite confused at this point

    Any help would be most appreciated and thanks in advanced
     
  2. jcsd
  3. Jan 26, 2014 #2

    ehild

    User Avatar
    Homework Helper
    Gold Member


    You have to show that x1,x2,x4 are linearly independent. Do the Gauss elimination with these three vectors.

    ehild
     
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