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Math Help - Help with a direct sum proof

  1. #1
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    Question Help with a direct sum proof

    The problem reads:

    "Let W1 denote the set of all polynomials f(x) in P(F) such that in the representation f(x) = (a_n)(x^n) + (a_(n-1))(x^(n-1)) +...+ (a_1)(x) + a_0, we have a_i = 0 whenever i is even. Likewise let W2 denote the set of all polynomials in g(x) in P(F) such that in the representation g(x) = (b_m)(x^m) + (b_(m-1))(x^(m-1)) +...+ (b_1)(x) + b_0, we have b_i = 0 whenever i is odd. Prove that P(F) = the direct sum of W1 and W2."


    Here's my thought's, but I do not know how to prove the 2nd axiom.

    So, W1 = all odd polynomials and W2 = all even polynomials.
    If P(F) = direct sum of W1 and W2,

    1. The intersection of W1 and W2 is zero.
    2. W1 + W2 = V

    #2. is obviously true since all odd polynomials plus all even polynomials is the set of all polynomials.

    However, does the intersection of W1 and W2 = 0?
    If not, then how can I form an example showing this?
    If so, how can I prove it?

    Please help me understand this!
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  2. #2
    Lord of certain Rings
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    Quote Originally Posted by very_unique View Post
    #2. is obviously true since all odd polynomials plus all even polynomials is the set of all polynomials.
    You have to prove it rigorously. If you know how to do it, then no problem.

    However, does the intersection of W1 and W2 = 0?
    If not, then how can I form an example showing this?
    If so, how can I prove it?

    Please help me understand this!
    If there exists a poly in the intersection, then both the even co-effs and the odd co-effs are 0. But this means all the co-effs are 0. Hence we are done
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  3. #3
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    I'm taking the first axiom too simply, then. What other things do I need to take into consideration?

    Isn't x^2 + x + 1 equal to the sum of x^2 + 1 (even function) and x (odd function)?
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  4. #4
    Lord of certain Rings
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    Quote Originally Posted by very_unique View Post
    I'm taking the first axiom too simply, then. What other things do I need to take into consideration?

    Isn't x^2 + x + 1 equal to the sum of x^2 + 1 (even function) and x (odd function)?
    You are right. I am just making sure you write the steps clearly

    It is just that you have to say/show the following:

    W_1 + W_2 \subset V: This follows trivially...
    V \subset W_1 + W_2: This one follows from what you said in the previous post...
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