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Isomorphism proof help/hint

  1. Mar 6, 2005 #1
    Let V denote the vector space that consists of all sequences {a_n} in F (field) that have only a finite number of nonzero terms a_n. Let W = P(F) (all polynomials with coefficients from field F). Define,

    T: V --> W by T(s) = sum(s(i)*x^i, 0, n)

    where n is the largest integer s.t. s(n) != 0. Prive that T is an isomorphism.


    I see how the transformation is mapping sequences to polynomials, but I don't even see how this is onto. Based on the sequence description, there comes a time where the remaining terms of every sequence is 0:

    s_n = (s1, s2, ..., sn, 0, 0, ...).

    So I don't see how that will "hit" every polynomial since the polynomials given in the problem don't have the "zero after finite many terms" restriction.
     
  2. jcsd
  3. Mar 6, 2005 #2

    mathwonk

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    gosh this is so obvious, what to say?

    you are just blocked from seeing it by some late night demon.

    Recall, what is the definition of a polynomial?


    In particular, what is the degree of a polynomial?

    or just try to produce ontoness directly by actually writing down say,

    the finite sequence that maps to the polynomial 5x^7 - x^6 +459x - 2.

    then ask yourself what prevents you from doing this for any polynomial.
     
    Last edited: Mar 6, 2005
  4. Mar 6, 2005 #3
    Is this right?

    For any p in P(F), p = c0 + c1 x + ... + ck x^k

    Then define T2 : W --> V by T2( c0 + c1 x + ... + ck x^k ) = (c0, c1, ...,
    ck, 0, 0, ...).

    Then,

    T2( T( (s0, s1, ..., sn, 0, 0, ...) ) )
    = T2( s0 + s1 x + ... + sn x^n )
    = (s0, s1, ..., sn, 0, 0, ...)

    and,

    T( T2( s0 + s1 x + ... + sn x^n ) )
    = T( (s0, s1, ..., sn, 0, 0, ...) )
    = s0 + s1 x + ... + sn x^n

    which implies T is invertible, and hence an isomorphism?
     
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