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Linear Independence of two functions and differentiation

  1. May 25, 2012 #1
    This is from my text, "Linear Algebra" by Serge Lang, pg 11:

    -The two functions et, e2t are linearly independent. To prove this, suppose that there are numbers a, b such that:

    aet + be2t=0

    (for all values of t). Differentiate this relation. We obtain

    aet + 2be2t = 0.

    Subtract the first from the second relation. We obtain be2t=0, and hence b=0. From teh first relation, it follows that aet=0, and hence a=0. Hence et, e2t are linearly independent.



    I'm confused as to the "Differentiate this relation". I see it creates a system of equations which can then be used to solve for linear independence, but why does it work?
     
  2. jcsd
  3. May 25, 2012 #2
    Hi srfriggen

    The first statement says that the function of t aet+b2t=0
    This must be true for all t, so this function of t is constant (always=0)
    therefore, if you look at its derivative, it must always be 0 too.
    So the second equation comes out, and since both equations are true, you can put them together and the answer comes out
     
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