Let $\{v_1, v_2, \ldots, v_n\}$ be an orthonormal basis for a finite-dimensional inner product space $V$ over some field $F$. For any $x, y$ in $V$, $\langle x, y \rangle = \sum\limits_{i = 1}^n \langle x, v_i \rangle \overline {\langle y, v_i \rangle}$. Prove that if $\beta$ is an orthonormal basis for $V$ with inner product $\langle\cdot, \cdot\rangle$, then for any $x,y \in V$, $\langle\phi_\beta(x), \phi_\beta(y)\rangle'=\langle[x]_\beta, [y]_\beta\rangle' = \langle x,y\rangle$, where $\langle \cdot, \cdot \rangle'$ denotes the standard inner product.
What's the general idea in proving this?