Adjoints and Inner Product Spaces

Question

The proposition states that if $V$ is a finite dimensional IPS and $\ \phi : V \rightarrow V$ is a linear operator, then $\phi$ has an adjoint $\phi^*$ and if the matrix of $\phi$ wrt to an orthonormal basis $B$ is $A$, then the matrix of $\phi^*$ wrt to $B$ is $A^{\dagger}$ (the complex conjugate of $A$).

The last part of the proof of this proposition is what I'm having trouble with - My notes state that since $\beta_{B} : \mathbb{R}^N$ or $\mathbb{C}^N \rightarrow V$ is an isomorphism (it doesn't state why this is - is it just a fact?)

$\phi^* := \beta_{B} \circ \phi_{A^{\dagger}} \circ \beta_{B}^{-1}$ $ \ $ satisfies $< \phi(\beta_{B}(x)) , \beta_{B}(y) >$ (where did this come from?) = $<\beta_{B}(x) , \phi^{*}(\beta_{B}(y))>$ (How is it equal to this?)

Perhaps I'm being incredibly slow but I really can't get my head around it so any help would be greatly appreciated.