Suppose we have the following linear application:
$ f : V \rightarrow W \\ \underline{x} \mapsto M(f)\underline{x}$
I know that there is an inverse linear application, $f^{-1}$ if and only if the linear application $f$ is biyective, which is proven in a very simple way.
Once I have purchased that it is invertible, my question is whether there is any relationship between the matrix associated with the application $f$, $M(f)$, and the matrix associated with the application $f^{-1}$, $M(f^{-1})$.
I, by mere intuition, believe that the relationship would be this one:
$M(f^{-1}) = (M(f))^{-1}$
Am I right? Or there is no relationship between the two matrices.
Yes, yo are right ! Suppose that $f$ is bijective and $f(x)=Mx$. Then, for some quadratic matrix $N$ we have $f^{-1}(y)=Ny$. Hence
$y=f(f^{-1}(y))=f(Ny)=MNy$ for all $y \in W$. In a similar fashion we get $x=NMx$ for all $x \in V$.
Thus: $MN=I_W$ and $NM=I_V$, therefore $N=M^{-1}$.