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Math Help - linearly independency in function space

  1. #1
    Member Jskid's Avatar
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    linearly independency in function space

    Let V be the vector space of all real-valued continuous functions. Is the following set linearly dependent? If yes express one vector as a linear combination of the rest.

    {cos(t), sin(t), e^t}

    I'm not sure how to get the equations to form a matrix, I don't see any homogeneous system here.
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  2. #2
    A Plied Mathematician
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    Try evaluating the Wronskian. What does that give you?
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  3. #3
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    Is your set of functions linearly dependent ?
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    Super Member girdav's Avatar
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    We can write a_1 \cos t+a_2\sin t+a_3e^t =0 and try to see if we must have a_1=a_2=a_3=0. What does this equation give if t=0? If t=\pi? If t=2\pi?
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  5. #5
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    \left ( \cos(t),\sin(t) \right ) is obviously l.independent.
    to prove that \left ( \cos(t),\sin(t),\exp(t) \right ) is dependent we must prove that
    \exp(t) \in \texttt{Span}(\cos(t),\sin(t)).
    Now suppose we have \exp(t) \in \texttt{Span}(\cos(t),\sin(t))
    \exp(t) \in \texttt{Span}(\cos(t),\sin(t))\Rightarrow \exp(t)=a\cos(t)+b\sin(t)
    for t=\frac{\pi}{2} we have b=\exp(\frac{\pi}{2})
    for t=0 we have a=1
    hence \exp(t)=\cos(t)+\exp(\frac{\pi}{2})\sin(t) which is not true.
    so \left ( \cos(t),\sin(t),\exp(t) \right ) must be l.independent.
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