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Math Help - Subspaces of polynomial vector space

  1. #1
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    Subspaces of polynomial vector space

    So I'm kind of running into a wall with this one. It should have pretty simple solutions but I'm just not seeing it.

    Let X be the the set of all polynomials with degree less than or equal to n. X is a vector space. A polynomial p e X has the form:

    p(x) = \sum_{k=1}^n a_kx^k

    Are the following sets subspaces of X for all x e R?

    W_1 = \{p \epsilon X | p(x) = p(-x)\}
    W_2 = \{p \epsilon X | p(x) = |p(x)|\}
    W_3 = \{p \epsilon X | p(x) = -p(-x)\}


    For W1 I've shown that:
    q_1 + q_2 = \sum_{k=1}^n (a_k + b_k)(-x)^k
    \lambda q_1 = \sum_{k=1}^n \lambda a_kx^k

    So it must be a subspace. But I'm not sure how to proceed with W2 and W3. Intuition tells me W2 is a subspace too, but I'm not sure how to show that mathematically. Any tips?
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  2. #2
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    Quote Originally Posted by cope View Post
    So I'm kind of running into a wall with this one. It should have pretty simple solutions but I'm just not seeing it.

    Let X be the the set of all polynomials with degree less than or equal to n. X is a vector space. A polynomial p e X has the form:

    p(x) = \sum_{k=1}^n a_kx^k

    Are the following sets subspaces of X for all x e R?

    W_1 = \{p \epsilon X | p(x) = p(-x)\}
    W_2 = \{p \epsilon X | p(x) = |p(x)|\}
    W_3 = \{p \epsilon X | p(x) = -p(-x)\}


    For W1 I've shown that:
    q_1 + q_2 = \sum_{k=1}^n (a_k + b_k)(-x)^k
    \lambda q_1 = \sum_{k=1}^n \lambda a_kx^k

    So it must be a subspace. But I'm not sure how to proceed with W2 and W3. Intuition tells me W2 is a subspace too, but I'm not sure how to show that mathematically. Any tips?


    p(x)=1\in W_2\,,\,\,but\,\,\,-p(x)\in W_2 ?

    W_3 is a subspace .

    Tonio
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  3. #3
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    Wow, that really was simple. And W3 is a subspace because we've already shown p(-x) to be a subspace, and -p(-x) just switches the signs of the coefficients. All coefficients from R are in W3, so -p(-x) must be in W3 too.

    Thanks
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