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Linearly independant vectors

  1. Apr 19, 2007 #1
    just a quick one:

    1. The problem statement, all variables and given/known data
    Show that the vectors a=2i -2j, b=3j - k and c = i + 2j +k are linearly independant


    2. Relevant equations



    3. The attempt at a solution

    What does 'linearly independent' mean and whats the test for it? Its from a really old exam paper so i might just know this theory under a different name.

    thankyou :smile:
     
  2. jcsd
  3. Apr 19, 2007 #2

    Dick

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    Science Advisor
    Homework Helper

    It means that the equation x*a+y*b+z*c=0 (where x,y,z are scalars and a,b,c are your vectors) has only the trivial solution x=y=z=0.
     
  4. Apr 20, 2007 #3
    Is this right then:

    (2i -2j)x + (3j - k)y + (i + 2j +k)z = 0

    multiply out and rearrange

    (2x + z)i + (-2x + 3y +2 z)j + (z - y)k = 0

    comparing is js and ks on each side

    2x + z = 0
    -2x + 3y + 2z = 0
    z - y = 0

    as matrices

    [2 0 1] [x] = [0]
    [-2 3 2] [y] = [0]
    [0 -1 1] [z] = [0]

    (like in the eigenvalue problem) there is a non-trivial solution only if determinent of the co-efficients is zero

    detM= 12

    =/= 0

    => vectors a,b,c where linearly independant
     
  5. Apr 20, 2007 #4

    radou

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    Looks right.
     
  6. Apr 22, 2007 #5
    thankyouuu
     
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