ועבור $w=w_j$,
$j=1,\dots,m$.
$$\left \langle T\left(v_i \right ),w_j \right \rangle=\left[T\left(v_i \right ) \right ]_{B'}^t\overline{\left[w_j \right ]_{B'}}=\left(A\left[v_i \right ]_B \right )^tA^t\overline{\left[w_j \right ]_{B'}}=e_i^tA^te_j=a_{ji}$$
$$\left \langle v_i,T^*\left(w_j \right ) \right \rangle=\left[v_i \right ]_B^t\overline{\left[T^*\left(w_j \right ) \right ]_B}=\left[v_i \right ]_B^t\overline{A'\left[w_j \right ]_{B'}}=e_i^t\overline{A'}e_j=\overline{a_{ij}'}$$