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Linear independence?

  1. Feb 18, 2009 #1
    Linear independence!?

    1. The problem statement, all variables and given/known data
    Let {p, q} be linearly independent polynomials. Show that {p, q, pq} is linearly independent if and only if deg(p)>=1 and deg(q)>=1.

    3. The attempt at a solution

    I am pretty sure the statement to prove is incorrect.
    If we use deg(p) = -1 and deg(q) = -2, we can easily show that the two are linearly independent (consider the functions p(x) = 1/x and q(x) = 1/x^2).
    We can have k/x + l/x^2 = 0
    then kx + l = 0.
    Then we can differentiate and get k = 0 and l = 0, which disproves the statement.
    How does this make any sense?
  2. jcsd
  3. Feb 18, 2009 #2


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    Re: Linear independence!?

    If you are talking about polynomials you are only talking about linear combinations of x^n where n>=0. There is so much more wrong with your counterexample, I don't know where to start... The statement is true, now try and prove it.
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