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Concept# Monomial basis

Summary

In mathematics the monomial basis of a polynomial ring is its basis (as a vector space or free module over the field or ring of coefficients) that consists of all monomials. The monomials form a basis because every polynomial may be uniquely written as a finite linear combination of monomials (this is an immediate consequence of the definition of a polynomial).
One indeterminate
The polynomial ring K[x] of univariate polynomials over a field K is a K-vector space, which has
1, x, x^2, x^3, \ldots
as an (infinite) basis. More generally, if K is a ring then K[x] is a free module which has the same basis.
The polynomials of degree at most d form also a vector space (or a free module in the case of a ring of coefficients), which has 1, x, x^2, \ldots as a basis.
The canonical form of a polynomial is its expression on this basis:
a_0 + a_1 x + a_2 x^2 + \dots + a_d x^d,
or, using the shorte

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