Proving Isomorphism between Vector Space and Polynomial Space

[eckiller]

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.

 

[mathwonk]

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.

 

[eckiller]

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?