# Monomial basis

In mathematics the monomial basis of a polynomial ring is its basis (as vector space or free module over the field or ring of coefficients) that consists in the set of all monomials. In fact, a polynomial may be uniquely written as a linear combination of monomials.

Univariate polynomials expressed on the monomial basis can be evaluated efficiently using Horner's method.

## Definition

The monomial basis for the vector space $\Pi_n$ of polynomials with degree n is the polynomial sequence of monomials

$1,x,x^2,.\ldots,x^n$

The monomial form of a polynomial $p \in \Pi_n$ is a linear combination of monomials

$a_0 1 + a_1 x + a_2 x^2 + \ldots + a_n x^n$

alternatively the shorter sigma notation can be used

$p=\sum_{\nu=0}^n a_{\nu}x^\nu$

## Notes

A polynomial can always be converted into monomial form by calculating its Taylor expansion around 0.

## Examples

A polynomial in $\Pi_4$

$1+x+3x^4$