Basis of Vector Space of Polynomial Functions

Theorem
Let $B$ be the set of all the identity functions $I^n$ on $\R^n$ where $n \in \N^*$.

Then $B$ is a basis of the $\R$-vector space $P \left({\R}\right)$ of all polynomial functions on $\R$.

Proof
By definition, every real polynomial function is a linear combination of $B$.

Suppose:
 * $\displaystyle \sum_{k \mathop = 0}^m \alpha_k I^k = 0, \alpha_m \ne 0$

Then by differentiating $m$ times, we obtain from Nth Derivative of Nth Power:
 * $m! \alpha_m = 0$

whence $\alpha_m = 0$ which is a contradiction.

Hence $B$ is linearly independent and therefore is a basis for $P \left({\R}\right)$.