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Math Help - linear independence of vector functions

  1. #1
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    linear independence of vector functions

    Hi all,

    I am given the question: Let I be an interval and let y1, . . . , yM be vector functions I - C^N. Show that if t0 I and y1(t), . . . , yM(t) C^N are linearly independent vectors then y1,....,ym are linearly independent functions.

    My attempt:

    Argue by contradiction. Thus assume the
    M functions y1(t), . . . , yM(t) are linearly dependent i.e.that there exist constants c1, . . . , cM not all zero such that

    \sum cjyj(t) = 0 for all t I. I want to evaluate at t=t0 but I'm not sure what to do next.



    Any suggestions would be awesome
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  2. #2
    MHF Contributor

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    Tejas
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    Re: linear independence of vector functions

    if the functions \{y_1,\dots,y_m\} are linearly independent, then there exists constants \{c_1,\dots,c_m\}

    such that \sum_{i=1}^m c_iy_i(t) = (0,\dots,0) for EVERY t in I (that is, the sum is the constant function which has the

    0-vector of \mathbb{C}^n as it's value for every t).

    but this means that for any t in I, \{y_1(t),\dots,y_m(t)\} is a linearly dependent set of vectors, the desired contradiction.
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