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Subspace Theorem

  1. Oct 9, 2008 #1
    Use the subspace Theorem to decide if the following are subspaces of P2, the vector space of all polynomials of degree at most 2.

    a) R1 = {ao + a1x +a2x^2 | ao = 0}
    b) R1 = {ao + a1x +a2x^2 | a1 = 1}
    c) R1 = { p E P2 | p has exactly degree 2}

    (for part c 'E' is 'element of')


    a) Is a subspace, closed under vector addition and scalar multiplication

    b) Isn't a subspace, vector addition doesn't hold take (a,1,c) + (d,1,e) = (a+d,2,c+e)
    the value for a1 is 1 so its not a subspace.

    c) Isn't a subspace, take -x^2 and x^2 under addition they equal 0 and aren't degree 2.

    I'm unsure of part c whether i've just interpreted it wrongly or just made a mistake but for some reason i just doesn't feel right..

    any help would be appreciated as usual, thanks :)
  2. jcsd
  3. Oct 9, 2008 #2
    Everything is correct from my POV.
  4. Oct 10, 2008 #3


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    Point of grammar, not mathematics: "and aren't degree 2" implies that -x^2 and x^2 are not of degree 2 which is, of course, not true. You should have said "which isn't degree 2" making it clear that it is their sum, the 0 polynomial, which is not of degree 2.

  5. Oct 10, 2008 #4
    Makes sense, will keep it in mind for future reference so i don't get arrested for bad grammar ;)

    Love you!
  6. Oct 10, 2008 #5


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    Staff Emeritus
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    You got a warning this time- next time a ticket!
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