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Help with figuring out Linear Dependency

  1. Oct 1, 2013 #1
    1. The problem statement, all variables and given/known data
    Hello, I have been given a list of functions and I need to figure out if they are linearly dependent or not. I've been trying to solve the problems for a while but I cannot figure out which ones are wrong. The list of functions are on my webpage at: http://vantraveller.blogspot.ca/2013/09/linear-dependency-question.html [Broken]


    2. Relevant equations

    http://vantraveller.blogspot.ca/2013/09/linear-dependency-question.html [Broken]

    3. The attempt at a solution
    For sure, I know cos^2(x) and sin^2(x), x and x^2, 1+ln(x) and 1-ln(x) is linearly independent because they are not multiples of each other, but with regards to the others, I am not to sure.
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
    Last edited by a moderator: May 6, 2017
  2. jcsd
  3. Oct 1, 2013 #2

    arildno

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    Set up the equation that must govern linear (in)dependency.
    Prove that the equation only holds when the coefficients for each function is set equal to zero.
     
  4. Oct 1, 2013 #3

    Mark44

    Staff: Mentor

    That's not a good reason. If you have two functions, it's easy to tell whether each is a multiple of the other, but with three or more functions, that thinking doesn't work any more. For example, {sin2(x), cos2(x), 1} is a linearly dependent set. No function is a multiple of any other in the set, but the equation c1 * sin2(x) + c2 * cos2(x) + c3 * 1 = 0 has a solution in which not all of the constants are zero.
     
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