- #1
eckiller
- 44
- 0
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.
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.