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Math Help - linearly independent

  1. #1
    Senior Member slevvio's Avatar
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    linearly independent

    Is the set  \{ \cos(nx) ,\sin(mx) \mid m,n \in \mathbb{N} \} \subseteq \{ f:\mathbb{R} \rightarrow \mathbb{R}\} linearly independent?

    #I have a vague kind of idea that those functions form a linearly independent set but only on [0,2pi]. Could anyone give me some advice on this? Thanks very much
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  2. #2
    MHF Contributor

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    Yes, they form a linearly independent set on the entire real line, not just [0, 2\pi]. Indeed, because they are periodic with period 2\pi, whatever is true of them on [0, 2\pi] is true for all x.
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  3. #3
    Senior Member slevvio's Avatar
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    Thanks! is there a way to prove that those functions are linearly independent in a simple way? It's easy to show if i take 2 functions out of it

    ie  \alpha \sin (nx) + \beta \cos (mx) = 0 \implies \alpha = \beta = 0 but I am having trouble doing it for an arbitrary subset
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