So $V$ is an inner product space (finite dimensional) with inner product defined.
If $v$ and $w$ are vectors in $V$, how would one go about proving this?
$\langle \phi_\beta (x), \phi_\beta (y) \rangle ' = \langle [x]_\beta, [y]_\beta \rangle ' = \langle x, y \rangle$
The $\langle \rangle '$ is the standard inner product on $F^n$. $\beta$ is an orthonormal basis for V.
Attempt: It seemed to me that the first half of the equality would be obvious, because $\phi_\beta (x) =[x]_\beta$ and $\phi_\beta (y) =[y]_\beta$, or at least that is how I know $\phi_\beta$ to be defined. I don't know if that's rigorous though.