Polynomial ring/Basis/Example

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

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

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

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

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