# Operations for Polynomials and Vector Space

In F[x] let V_n be the set of all polynomials of degree less than n. Using the natural operations for polynomials of addition and multiplication, V_n is a vector space over F.

Any element of V_n is of the form a_0 + a_1x + a_2x^2 + ... + a_(n-1)x^(n-1) where a_i belongs to F.

Let F be a field and let V be the totality of all ordered n-tuples, (a_0, a_1, a_2, ... a_n) where the a_i belongs to F.

Two elements (a_0, a_1, a_2, ... a_n) and (b_0, b_1, b_2, ... b_n) of V are declared to be equal if and only if a_i = b_i for each i = 1,2,..., n.

The operations in V to make of it a vector space is defined as

(1) (a_0, a_1, a_2, ... a_n) + (b_0, b_1, b_2, ... ,b_n) = (a_0 + b_0 ,a_1 + b_1 ,a_2 + b_2, ..., a_n + b_n )

(2) r(a_0, a_1, a_2, ... a_n) = (ra_0, ra_1, ra_2, ..., ra_n) for r belongs to F.

V is a vector space over F, we assign a symbol to it, namely F(n).

Prove that the vector space Vn is isomorphic to the vector space F(n).

See the attached file.

#### Solution Summary

This solution is comprised of a detailed explanation of the two vector spaces given below:

In F[x] let V_n be the set of all polynomials of degree less than n. Using the natural operations for polynomials of addition and multiplication, V_n is a vector space over F.

Any element of V_n is of the form a_0 + a_1x + a_2x^2 + ... + a_(n-1)x^(n-1) where a_i belongs to F.

Let F be a field and let V be the totality of all ordered n-tuples, (a_0, a_1, a_2, ... a_n) where the a_i belongs to F.

Two elements (a_0, a_1, a_2, ... a_n) and (b_0, b_1, b_2, ... b_n) of V are declared to be equal if and only if a_i = b_i for each i = 1,2,..., n.

The operations in V to make of it a vector space is defined as

(1) (a_0, a_1, a_2, ... a_n) + (b_0, b_1, b_2, ... ,b_n) = (a_0 + b_0 ,a_1 + b_1 ,a_2 + b_2, ..., a_n + b_n )

(2) r(a_0, a_1, a_2, ... a_n) = (ra_0, ra_1, ra_2, ..., ra_n) for r belongs to F.

V is a vector space over F, we assign a symbol to it, namely F(n).

It contains step-by-step explanation to prove that the vector space V_n is isomorphic to the vector space F(n).

Notes are also given at the end.

Solution contains detailed step-by-step explanation.

Notes are also given at the end.