What do we understand when we say the matrix of the linear operator $A $ in the basis $ E $ is $ M $?
Acording to the definition
$ A: L_1 \rightarrow L_2 $
$M: V_n(K) \rightarrow V_M(K) $
the basis of $\ L_1 $ is $\ E_1 $ and the basis of $ L_2 $ is $ E_2 $. Then $ AE_1=ME_2 $ and here $ M $ is the matrix of the linear operator $ A $ in the bases $ E_1 $ and $ E_2 $.