Orthonormal basis, isomorphism preserving dot product

Question

For an orthonormal basis $v_1, ..., v_n$ of $(V, \cdot )$

$\mathbb{R}^n \ni (x_1, ..., x_n) \rightarrow \sum x_jv_j \in V$

is an isomorphism preserving dot product.

I've already proven that it preserves dot product, but I have problems showing that it is isomorphic.

Could you tell me how to do it?

Answer 1

The linear map $\phi_v:\mathbb R^n\rightarrow V$ is s.t. $\phi_v(x_1,\dots,x_n):=\sum_{i=1}^n x_iv_i$, denoting by $v$ the orthonormal basis $\{v_1,\dots, v_n\}$ in $V$. It admits inverse given by $\psi:V\rightarrow \mathbb R^n$, with $\psi(w)=(w_1,\dots,w_n)$, for any $w\in V$, and $w=\sum_{i=1}^n w_iv_i$.