Equivalent matrices represent the same linear map in different bases It is clear from the change of basis formula that the matrices $A$ and $B$ representing the same linear map in different bases are equivalent: there exists invertible matrixes $Q$ and $P$ such that $A=Q^{-1}BP$.
My question is about the other way, I mean given two equivalent matrices $A$ and $B$ in $M_{n,p}(\mathbb K)$ why they must represent the same linear map in different bases ? thank you for your help!

Clearly $B$ being an $m \times n$ matrix represents a transformation $\Bbb R^n \rightarrow \Bbb R^m$ where we take the standard basis in each space.  If $A = Q^{-1}BP$, then take a basis for $\Bbb R^n$ made up of the columns of $P$ and a basis for $\Bbb R^m$ made up of the columns of $Q$.  Then $A$ represents the transformation with respect to these new bases.
