monomial basis

In mathematics a monomial basis is a way to uniquely describe 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.

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 converted into monomial form by calculating Taylor expansion around 0.

Examples

A polynomial in

\Pi_4

1+x+3x^4

Monomial Basis

Definition

Notes

Examples

See also