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Subspace of P4

  1. Feb 25, 2008 #1
    1. The problem statement, all variables and given/known data

    Determine whether the following is a subspace of [tex]P_{4}_[/tex]

    (a) The set of polynomials in [tex]P_{4}[/tex] of even degree.

    2. Relevant equations

    [tex]P_{4} = ax^{3}+bx^{2}+cx+d[/tex]

    3. The attempt at a solution

    [tex](p+q(x)) = p(x) + q(x)[/tex]
    [tex](\alpha p)(x) = p(\alpha x)[/tex]

    If p and q are both of degree 2 then both scalar multiplication and vector addition should return a polynomial of degree 2 as far as I can tell, however my book states that this is not a subspace of [tex]P_{4}[/tex]. I can't tell why it wouldn't be.
     
  2. jcsd
  3. Feb 25, 2008 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    The set of "polynomials of degree 2" are specifically those of the form ax2+ bx+ c with a non-zero. In particular, that does not include the 0 vector. Further, if p1(x)= x2+ 2x+ 1 and p2(x)= -ax2+ 3x+ 1, thenthe sum is NOT a "polynomial of degree 2". The polynomials in P4 that are of even degree are either of degree 2 or degree 0. In any case, the example I just gave answers your question.
     
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