Polynomial ring/Basis/Example

In the polynomial ring ${}K[X]$ over a field ${}K$, the powers ${}X^{n}$, ${}n\in \mathbb {N}$, form a basis. By definition, every polynomial

$a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{2}X^{2}+a_{1}X+a_{0}$

is a linear combination of the powers ${}X^{0}=1,X^{1}=X,\ldots ,X^{n}$. Moreover, these powers are linearly independent. For if

${}a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{2}X^{2}+a_{1}X+a_{0}=0\,,$

then all coefficients equal ${}0$ (this is part of the concept of a polynomial).