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Basis Proof

  1. May 5, 2009 #1
    1. The problem statement, all variables and given/known data

    Prove (u,v,u+v) can not be a basis for <u,v,u+v>.


    2. Relevant equations



    3. The attempt at a solution

    Let αu+βv+γ(u+v)=0
    αu+βv=-γ(u+v)
    α/γ(u)+β/γ(v)=-(u+v)
    α/γ(u)+β/γ(v)+1*(u+v)=0

    Since α/γ,β/γ,1 are not all zeros, therefore, (u,v,u+v) are linearly dependent. Hence it doesn't form basis for <u,v,u+v>.


    Let me know if this is the right approach towards the proof.
     
  2. jcsd
  3. May 5, 2009 #2
    Showing that (u,v,u+v) are linearly dependent is the correct way to prove they don't form a basis. To show that they are linearly dependent, you only have to find an example of α, β, and γ (not all zero) for which αu+βv+γ(u+v)=0. I would just write down a specific choice of numbers that works; this is not difficult to do by guess and check.
     
  4. May 5, 2009 #3
    So I guess we can achieve that by saying the following:

    -1*(u) + (-1)*v + 1*(u+v) = 0

    Since none of the constants are actually zero. Therefore, (u,v,u+v) are infact linealy dependent.
     
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