A-10.8 Transforming Basis between Polynomial Ring and Vector Space

Suppose some polynomials $f_0(X),f_1(X),\cdots ,f_{n-1}(X)$ form a basis of the $(n-1)$-degree polynomial ring and $\sigma$ is an isomorphic mapping from the $(n-1)$-degree polynomial ring to the $n$-dimensional vector space $\in {\mathbb{Z}}^n$. Then, $(\sigma (f_0(X)),\sigma (f_1(X)),\cdots ,\sigma (f_{n-1}(X)))$ form a basis of the $n$-dimensional vector space. This is because the $\sigma$-mapped output vectors homomorphically preserve the same algebraic relationships on the $\text{(+,⋅)}$ operations and the basis relationship between basis vectors and a subspace can be expressed as a linear algebraic formula consisting of the $\text{(+,⋅)}$ operations (i.e., linear independence and spanning of the space). Therefore, if a set of polynomials satisfies a basis relationship, their $\sigma$-mapped vectors also preserve a basis relationship.

The same principle holds between a polynomial ring and vector space over complex numbers. Given the polynomial ring $ℝ[X]∕(x^n+1)$, the most intuitive way to set up a basis of $ℝ[X]∕(x^n+1)$ is as follows:

$f_0(X)=1$

$f_1(X)=X$

$f_2(X)=X^2$

$⋮$

$f_{n-1}(X)=X^{n-1}$

These $n$ polynomials are linearly independent, because each polynomial exclusively has its own unique exponent term, whereas one term cannot be expressed by a linear combination of the other terms. Also, these $n$ polynomials span the polynomial ring $ℝ[X]∕(x^n+1)$, because each polynomial’s scalar multiplication can express any coefficient value of its own exponent term, and summing all such polynomials can express any polynomial in the polynomial ring $ℝ[X]∕(X^n+1)$.

Now, we will apply the ${\sigma }_c$ mapping to the above $n$ polynomials that are a basis of the $(n-1)$-degree polynomial ring $ℝ[X]∕(x^n+1)$. Then, according to the principle of polynomial-to-vector basis transfer (explained in Theorem A-10.5 in §A-10.5), we can use these $n$ polynomials (i.e., the basis of the $(n-1)$-degree polynomial ring) and the isomorphic polynomial-to-vector mapping ${\sigma }_c$ to compute the basis of the $n$-dimensional special vector space ${\widehat{ℂ}}^n$ as follows:

$W=\left[\begin{array}{c}\hfill {\sigma }_c(f_0(X))\hfill \\ \hfill {\sigma }_c(f_1(X))\hfill \\ \hfill {\sigma }_c(f_2(X))\hfill \\ \hfill ⋮\hfill \\ \hfill {\sigma }_c(f_{n-1}(X))\hfill \end{array}\right]=\left[\begin{array}{c}\hfill {\sigma }_c(1)\hfill \\ \hfill {\sigma }_c(X)\hfill \\ \hfill {\sigma }_c(X^2)\hfill \\ \hfill ⋮\hfill \\ \hfill {\sigma }_c(X^{n-1})\hfill \end{array}\right]$ $=\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill & \hfill 1\hfill \\ \hfill (\omega )\hfill & \hfill ({\omega }^3)\hfill & \hfill ({\omega }^5)\hfill & \hfill \cdots \hfill & \hfill ({\omega }^{2n-1})\hfill \\ \hfill {(\omega )}^2\hfill & \hfill {({\omega }^3)}^2\hfill & \hfill {({\omega }^5)}^2\hfill & \hfill \cdots \hfill & \hfill {({\omega }^{2n-1})}^2\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {(\omega )}^{n-1}\hfill & \hfill {({\omega }^3)}^{n-1}\hfill & \hfill {({\omega }^5)}^{n-1}\hfill & \hfill \cdots \hfill & \hfill {({\omega }^{2n-1})}^{n-1}\hfill \end{array}\right]$

$=\left[\begin{array}{ccccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill \cdots \hfill & \hfill 1\hfill \\ \hfill (\omega )\hfill & \hfill ({\omega }^3)\hfill & \hfill \cdots \hfill & \hfill {(\overline{\omega })}^3\hfill & \hfill (\overline{\omega })\hfill \\ \hfill {(\omega )}^2\hfill & \hfill {({\omega }^3)}^2\hfill & \hfill \cdots \hfill & \hfill {({\overline{\omega }}^3)}^2\hfill & \hfill {(\overline{\omega })}^2\hfill \\ \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill ⋮\hfill & \hfill \ddots \hfill & \hfill ⋮\hfill \\ \hfill {(\omega )}^{n-1}\hfill & \hfill {({\omega }^3)}^{n-1}\hfill & \hfill \cdots \hfill & \hfill {({\overline{\omega }}^3)}^{n-1}\hfill & \hfill {(\overline{\omega })}^{n-1}\hfill \end{array}\right]$

$W$ is a valid basis of the $n$-dimensional special vector space ${\widehat{ℂ}}^n$.