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 $B$.
See Polynomial Function for where this is actually documented. Work in progress to merge this with the definition.
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Suppose:
- $\displaystyle \sum_{k=0}^m \alpha_k I^k = 0, \alpha_m \ne 0$
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 $P \left({\R}\right)$.
$\blacksquare$
Sources
- Seth Warner: Modern Algebra (1965): $\S 27$: Example $27.7$
