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Abstract Linear Algebra, Linear Functional

  1. Sep 24, 2012 #1

    chm

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    1. The problem statement, all variables and given/known data
    problem didn't state, but I assume let V be a vector space: V = C^3 and scalar is C


    2. Relevant equations
    Define a non-zero linear functional T on C^3 such that T ((1, 1, 1)) = T ((1, 1, −1)) = 0


    3. The attempt at a solution
    So let X1 = (1, 1, 1); X2 = (1, 1, -1);
    It asks me to define a non zero linear functional, that means I need to define T(x) = α1T(x1) + α2T(X2) ?
    Since T(X1) = T(X2) = 0 then T(X) = 0 then it's not the answer.
    I'm stuck here. Can anyone give me a hint please? Thanks!
     
  2. jcsd
  3. Sep 24, 2012 #2

    Dick

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    Homework Helper

    T((x,y,z))=T((x,0,0))+T((0,y,0))+T((0,0,z))=x*T((1,0,0))+y*T((0,1,0))+z*T((0,0,1)). Because it's a linear functional, yes? Now you just need to pick values for T((1,0,0)), T((0,1,0)) and T((0,0,1)) that give the same value of 0 for your input vectors. Many choices are possible, just pick one.
     
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