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 polynomial function is a linear combination of $$B$$.

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

Then by differentiating $$m$$ times, we obtain:
 * $$m! \alpha_m = 0$$

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

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