Change of Basis is Invertible

Theorem
Let $R$ be a ring with unity.

Let $M$ be a free $R$-module of finite dimension $n>0$.

Let $\mathcal A$ and $\mathcal B$ be ordered bases of $M$.

Let $\mathbf P$ be the change of basis matrix from $\mathcal A$ to $\mathcal B$.

Then $\mathbf P$ is invertible, and its inverse $\mathbf P^{-1}$ is the change of basis matrix from $\mathcal B$ to $\mathcal A$.

Proof
From Product of Change of Basis Matrices and Change of Basis Matrix Between Equal Bases:


 * $\left[{I_M; \mathcal A, \mathcal B}\right] \left[{I_M; \mathcal B, \mathcal A}\right] = \left[{I_M; \mathcal A, \mathcal A}\right] = I_n$


 * $\left[{I_M; \mathcal B, \mathcal A}\right] \left[{I_M; \mathcal A, \mathcal B}\right] = \left[{I_M; \mathcal B, \mathcal B}\right] = I_n$

Hence the result.