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Linear dependence of non-numerical objects

  1. Oct 23, 2012 #1
    1. The problem statement, all variables and given/known data

    Suppose V = 2^Ω where Ω = {red,blue,yellow,green}. Verify whether u={blue,green}, v={red,yellow,green}, and w={blue,yellow} are linearly independent in V


    3. The attempt at a solution

    I let
    red = a1
    blue = a2
    yellow =a3
    green = a4

    therefore Ω = {a1, a2, a3, a4}

    so now u={a2,a4}, v={a1,a3,a4}, w={a2,a3}

    These vectors can also be written as
    u=0.a1+1.a2+0.a3+1.a4
    v=1.a1+0.a2+1.a3+1.a4
    w=0.a1+1.a2+1.a3+0.a4

    writing these vectors out in one matrix gives us
    (0 | 1 | 0
    1 | 0 | 1
    0 | 1 | 1
    1 | 1 | 0) if we're supposed to write the vectorsas columns

    then i reduced it to REF to get
    (1 | 0 | 0
    0 | 1 | 0
    0 | 0 | 1
    0 | 0 | 0)

    But then what do I do? How do I know what is dependent or not? I thought I could just say a1=0, a2=0,a3=0,a4=free. That shows these are independent but there is an empty row surely something's dependent??
     
  2. jcsd
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