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Vector space of polynomials problem

  1. Oct 29, 2007 #1
    1. Consider the vector space of polynomials 1+x^3 , 1-x+x^2, 2x, 1+x^2
    Are they linearly dependent or independent? dimension of vecotr space spanned by these vectors?

    3. I have tried to solve this by letting
    a1 = 1+x^3
    a2 = 1-x+x^2
    a3 = 2x
    a4 = 1+x^2

    Then I let
    (alpha)a1 + (beta)a2 + (gamma)a3 + (delta)a4 = 0
    (alpha)(1+x^3) + (beta)(1-x+x^2) + (gamma)(2x) +(delta)(1+x^2) = 0
    (alpha +beta+delta) + x(2gamma - beta) + x^2(beta + delta) + x^3(alpha) = 0

    So alpha + beta+ delta = 0
    2gamma - beta = 0
    beta + delta = 0
    alpha = 0

    But I can't solve for beta delta and gamma so how do I know if their independent or dependent?
  2. jcsd
  3. Oct 29, 2007 #2


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    There's no unique solution, so you'll have to write the solution in terms of some of your other variables. In this case it looks easy to solve for the other variables in terms of beta. Then ask yourself if there are any nonzero solutions.
  4. Oct 29, 2007 #3

    In that case beta = -delta
    and gamma = beta/2
    and alpha = 0

    How can you tell if they are non-zero?
    How do you find the dimension?
  5. Oct 29, 2007 #4


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    That gives you a solution for every value of beta. So putting beta=1 tells you what about linear independence?
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