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Linear independence of the set of exponential functions

  1. Sep 21, 2008 #1
    1. The problem statement, all variables and given/known data
    For each [tex] n \in \mathbb{N},[/tex] let [tex]f_n(x) = e^{nx}[/tex] for [tex]x \in \mathbb{R}[/tex]. Prove that [tex] f_1, ... , f_n [/tex] are linearly independent vectors in [tex] {\cal F}(\mathbb{R}, \mathbb{R})[/tex]

    2. Relevant equations

    3. The attempt at a solution
    I know that the simple way to prove this for n=2 would be by setting x to 0 and 1 and showing that c_1 and c_2 must be 0 with two simultaneous equations. However I don't know how to generalize that to an arbitrary n. I considered making a generalized Wronskian, but I think that would get sloppy and confusing very quickly.
  2. jcsd
  3. Sep 21, 2008 #2


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    What happens when you differentiate both sides of

    [tex]c_1 f_1 + \cdots + c_n f_n = 0?[/tex]
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