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Math Help - Linear dependence

  1. #1
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    Linear dependence

    hi


    Determine whether the functions in the following set are linearly dependent or linearly independent. If they are linearly dependent find a linear equation which they satisfy.



    { ln(x - 1), 2ln(x + 1), 3ln(x^2 - 1) } for x > 2


    i know the Wronskian is W(ln(x - 1), 2ln(x + 1), 3ln(x^2 - 1))

    = 12ln(x^2 - 1)/((x - 1)^2(x + 1)^2) - 12ln(x + 1)/((x - 1)^2(x + 1)^2)

    - 12ln(x - 1)/((x - 1)^2(x + 1)^2)


    and if i put x = 5 say, then this equals zero, so equations are lin dependent for some interval I.


    My problem is finding a linear equation which they satisfy!


    I understand that

    12ln(x^2 - 1)/((x - 1)^2(x + 1)^2) - 12ln(x + 1)/((x - 1)^2(x + 1)^2)

    - 12ln(x - 1)/((x - 1)^2(x + 1)^2) = 0

    how do i explain the rest of the question??


    help appreciated
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  2. #2
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    I found the Wronskian to be:

    \frac{12ln(x^{2}-1)}{(x-1)^{2}(x+1)^{2}}-\frac{12ln(x+1)}{(x-1)^{2}(x+1)^{2}}-\frac{12ln(x-1)}{(x-1)^{2}(x+1)^{2}}

    This equals 0 for x\geq{2}

    The function is not 0 over (-\infty,\infty), so it must be
    linearly independent. Correct?.

    Here's an interesting example of an linearly dependent set we can find by using identities.

    sin^{2}(x), \;\ cos^{2}(x), \;\ 5

    Since 5sin^{2}(x)+5cos^{2}(x)-5=5(sin^{2}(x)+cos^{2}(x))=-5=0

    This equals 0 for all x. So, it's linearly dependent.
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  3. #3
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    Wink

    hi

    thanks for the insight, i'm gonna have to get to grips with this, may come up on the exam, i am just struggling with the;

    If they are linearly dependent (which they are!) find a linear equation which they satisfy part of the question!


    ta
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