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.

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# Isomorphism proof help/hint

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