Prove that $h:\mathbb{R}^n \to \mathbb{R}^n $ (not necessarily linear) can be written uniquely of the form $h(x) = \sum_{i=1}^n \alpha_i (x) a_i,$

Question

In the book of Analysis on Manifolds by Munkres, at page 175, it is given that

Let $h:\mathbb{R}^n \to \mathbb{R}^n $ be a map s.t $h(0) = 0$. [...] Assume that $h$ is an isometry with $h(e_i) = a_i$ for all $i$.Then for each $x \in \mathbb{R}^n$, $h(x)$ can be written uniquely in the form $$h(x) = \sum_{i=1}^n \alpha_i (x) a_i,$$ for certain real-valued functions $\alpha_i (x)$ of $x$.

However he does not give any justification for the existence & uniqueness of the functions $\alpha_i$s, so my question is that

How can we show the existence and the uniqueness of $\alpha_i$s (the functions)?

Answer 1

This is just the fact that the $a_i$ are a basis of $\mathbb R^n$. For that you can use the "isometry" property.