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Show that two sets of vectors span the same subspace

  1. Dec 14, 2011 #1
    1. The problem statement, all variables and given/known data
    Show that the two sets of vectors
    {A=(1,1,0), B=(0,0,1)}
    and
    {C=(1,1,1), D=(-1,-1,1)}
    span the same subspace of R3.


    2. Relevant equations
    {A=(1,1,0), B=(0,0,1)}
    {C=(1,1,1), D=(-1,-1,1)}

    3. The attempt at a solution
    aA+bB=(a,a,0)+(0,0,b)=(a,a,b)
    aC+bD=(a,a,a)+(-b,-b,b)=(a-b,a-b,a+b)
    I am confused because I thought the answers would turn out to be equal.. Is this not the way to do it?
     
  2. jcsd
  3. Dec 14, 2011 #2

    radou

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    Homework Helper

    Well, take (a-b, a-b, a+b). Now, this equals a(1, 1, 1) + b(-1, -1, 1). Of what form are the vectors (1, 1, 1) and (-1, -1, 1)? Do they look familiar?
     
  4. Dec 14, 2011 #3
    its aC+bD... i don't understand how this is related to aA+bB??
     
  5. Dec 14, 2011 #4

    radou

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    Homework Helper

    The two vectors I mentioned are in your space "(a, a, b)". Hence, their linear combination is in that space, too.
     
  6. Dec 14, 2011 #5
    Oh! Thank you! now i see it lol
     
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