Q:

Similar Matrices and Linear Transformations

There is a theorem in the course notes but the lecturer didn't given the proof: 
If $A,B\in M_n(K)$, where $K$ is a field. Then $A\sim B$ if and only if they represent the same linear transformation $T$ of an $n$-dimensional vector space $V$ over $K$ with respect to bases $\mathcal{B}$ and $\mathcal{C}$. i.e., 
$$
A\sim B\iff A=[T]_\mathcal{B},\ B=[T]_{\mathcal{C}}.
$$
I know how to prove the '$\Leftarrow$' but I dont think I can prove the other direction '$\Rightarrow$'. 
Could please somebody help me?

A:

$A\sim B$ means that there exists an invertible matrix $P$ such that $B=P^{-1}AP$. An invertible matrix $P$ can be considered as the change of basis matrix $P=[I]^E_C$ from the standard basis $E$ of $K^n$ to the basis $C$ of consisting of the columns of the matrix $P$. Thus if you define the linear transformation
$$T:K^n\to K^n$$
by $$T(v)=B\cdot v$$ you will get that $[T]_E=B$ and $[T]_C=[I]^E_C\cdot B\cdot [I]^C_E=PBP^{-1}=A$.

