Change of Basis Matrix from Basis to Itself is Identity

Theorem
Let $R$ be a ring with unity.

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

Let $\mathcal B$ be an ordered basis of $M$.

Then the change of basis matrix from $\mathcal B$ to $\mathcal B$ is the $n\times n$ identity matrix:
 * $\mathbf M_{\mathcal B, \mathcal B} = \mathbf I$

Proof
Follows directly from the definition of change of basis matrix.