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,(adsbygoogle = window.adsbygoogle || []).push({});

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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Isomorphism proof help/hint

**Physics Forums | Science Articles, Homework Help, Discussion**