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Linear Dependence/Independence and Wronskian

  1. Sep 8, 2015 #1
    So I know there are a few threads and many websites on this, but I am not finding what I am looking for.

    To determine whether a set of functions are linearly dependent or independent I understand that the Wronskian can be used, but many example problems state that "clearly by inspection" some functions are dependent or independent.

    This method of inspection is not always trivial for me, so is the Wronskian a guaranteed way to solve these problems?

    For example if:

    [itex]f = x,[/itex] [itex]g = x+2,[/itex] [itex]h = x+5[/itex]

    The Wronskian generates a zero which shows dependence. Upon inspection I would think otherwise, but then again I am not a mathematician.
     
  2. jcsd
  3. Sep 8, 2015 #2

    RUber

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    2h - 5g + 3f = 0. Therefore they are dependent.
    As for whether the wronskian is a guaranteed method, I will refer to: http://mathwiki.ucdavis.edu/Analysi...uations/Linear_Independence_and_the_Wronskian

    Which provides the theorem:
    Let f and g be differentiable on [a,b] . If Wronskian W(f,g)(t_0 ) is nonzero for some t_0 in [a,b] then f and g are linearly independent on [a,b]. If f and g are linearly dependent then the Wronskian is zero for all t in [a,b] .
     
  4. Sep 8, 2015 #3

    DEvens

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  5. Sep 8, 2015 #4
    Thank you both, perhaps my example was too trivial, but nevertheless I greatly appreciate your responses.
     
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