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Linear independence problem

  1. Oct 17, 2006 #1
    I'm stuck on a question in linear algebra, it reads "Show that the subset S={cos mx, sin nx: m between 0 and infinity, n between 1 and infinity} is linearly independent.

    I really just don't know where to start. I've seen a similar question which was just sin (nx) and the lecturer integrated sin(px)sin(qx) between -pi and pi but I just don't see why he did that or anything.

    Any starting help would be much appreciated, thanks.
  2. jcsd
  3. Oct 17, 2006 #2


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    My guess is he showed that sin(px) and sin(qx) are orthogonal if p/=q. If two nonzero vectors are orthogonal then they are linearly independent.
  4. Oct 17, 2006 #3


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    The definition of "linearly independent", applied here would be that
    [itex]a_0+ a_1cos(x)+ b_1sin(x)+ a_2cos(2x)+ b_2sin(2x)+ ...= 0[/itex] only when each [itex]a_i[/itex] and [itex]b_i[/itex] is 0. What would you get if you multiply that sum by sin(nx) or cos(nx), for all n, and integrate?
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