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More on vector spaces

  1. Feb 21, 2006 #1
    Determine if this is a vector space with the indicated operations

    the set of V of all polynominals of degree >=3, togehter iwth 0, operations of P (P the set of polynomials)

    now all the scalar multiplication axioms hold.
    the text however says that the axion
    [tex] \mbox{For u,v} \in V, \mbox{then} \ u+v \in V [/tex] does not hold

    well ok take two polynomials
    [tex] u(x) = a_{3} x^3 + ... + a_{n} x^n [/tex]
    [tex] v(x) = b_{3} x^3 + ... + b_{k} x^k [/tex]
    where both n,k>= 3, then suppose k< n
    [tex] u(x) + v(x) = (a_{3} + b_{3}) x^3 + ... + (a_{k} + b_{k}) x^k + ... + a_{n} x^n [/tex]
    which is certainly a polynomial or degree >= 3 isnt it?
    It also applies for n<k and n = k
    is the text book wrong?
  2. jcsd
  3. Feb 21, 2006 #2

    matt grime

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    What is the degree of the following poly


    now, do you see your error?
  4. Feb 21, 2006 #3
    ok let me correct that then
    n,k >= 3
    [tex] u(x) = a_{0} + a_{1} x + ... + a_{n} x^n [/tex]
    [tex] v(x) = b_{0} + b_{1} x + ... + b_{k} x^k [/tex]
    then for n< k
    [tex] u + v = (a_{0} + b_{0}) + ... + (a_{k} + b_{k}) x^k + ... + a_{n} x^n [/tex]

    stil lseems to be of degree three to me
    however if k=n and an= -bn then the polynomial is no more degree 3
    is this corret?
  5. Feb 21, 2006 #4

    matt grime

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    Why don't you just find a counter example? two polys of degree 3 or greaterwhose sum isn't? A single counter examplem suffices.
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