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Linear independence

  1. Sep 28, 2011 #1
    1. The problem statement, all variables and given/known data

    there is the vector space F(R) = {f | f:R -> R }
    show that {1, sin^2(x), sin(2x)} is linearly independent



    2. Relevant equations

    a(1) + b(sin^2(x)) + c(sin(2x)) = 0, where the ONLY solution is a=b=c=0, for the set to be implied linearly independent.


    3. The attempt at a solution

    for that set to be considered linearly independent, it has to be linearly independent (a=b=c=0) for ALL values of x?

    i mean, for x = 0

    a(1) + b(sin^2(x)) + c(sin(2x)) = 0

    0(1) + 1(0) + 1(0) = 0, and that would be a linearly dependent set since not all coefficients are 0.

    but that is only one case. do i have to show that this is not valid for EVERY case? what would be a good way to approach these types of problems?

    Thanks!
     
  2. jcsd
  3. Sep 28, 2011 #2

    Mark44

    Staff: Mentor

    Yes. For the three functions to be linearly independent, the equation a(1) + b(sin^2(x)) + c(sin(2x)) = 0 hold for all values of x, and the only solutions for the constants must be a = b = c = 0.
     
  4. Sep 28, 2011 #3
    what would be a valid way to show that the set is linearly independent?
     
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