Matrix_Corresponding_to_Change_of_Basis_under_Linear_Transformation

Let $R$ be a ring with unity. Let $G$ and $H$ be free $R$-modules of finite dimensions $n,m>0$ respectively. Let $\left \langle {a_n} \right \rangle$ and $\left \langle {{a_n}'} \right \rangle$ be ordered bases of $G$. Let $\left \langle {b_m} \right \rangle$ and $\left \langle {{b_m}'} \right \rangle$ be ordered bases of $H$. Let $u: G \to H$ be a linear transformation, and let $\left[{u; \left \langle {b_m} \right \rangle, \left \langle {a_n} \right \rangle}\right]$ be the matrix of $u$ relative to $\left \langle {a_n} \right \rangle$ and $\left \langle {b_m} \right \rangle$. Let $\mathbf A = \left[{u; \left \langle {b_m} \right \rangle, \left \langle {a_n} \right \rangle}\right]$ $\mathbf B = \left[{u; \left \langle {{b_m}'} \right \rangle, \left \langle {{a_n}'} \right \rangle}\right]$ Then $\mathbf B = \mathbf Q^{-1} \mathbf A \mathbf P$ where $\mathbf P$ is the matrix corresponding to the change of basis from $\left \langle {a_n} \right \rangle$ to $\left \langle {{a_n}'} \right \rangle$ $\mathbf Q$ is the matrix corresponding to the change of basis from $\left \langle {b_m} \right \rangle$ to $\left \langle {{b_m}'} \right \rangle$.