Let $k$ be a field, write $0_k$ for the additive identity of $k$, let $d \in \mathbb{N} \cup \{0\}$, and let $x$ denote the indeterminate in the [polynomial ring](/page/Polynomial%20Ring) $k[x]$. Define
paragraph
admin
\begin{align*}
\mathcal{P}_d(k) := \left\{p \in k[x] : p=\sum_{i=0}^{d} a_i x^i \text{ for some } a_0,\dots,a_d \in k\right\}.
\end{align*}
latex_env
admin
Then $\mathcal{P}_d(k)$ is a [vector space](/page/Vector%20Space) over $k$ under the addition and scalar multiplication inherited from $k[x]$. Moreover, the ordered list
paragraph
admin
\begin{align*}
1,x,x^2,\dots,x^d
\end{align*}
latex_env
admin
is a basis of $\mathcal{P}_d(k)$ over $k$, and hence