Monomial basis

Monomial basis

In mathematics a monomial basis is a way to describe uniquely a polynomial using a linear combination of monomials. This description, the monomial form of a polynomial, is often used because of the simple structure of the monomial basis.

Polynomials in monomial form can be evaluated efficiently using the Horner algorithm.

Contents

Definition

The monomial basis for the vector space Πn 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 Π4

1 + x + 3x4

See also


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