Change of Basis Matrix from Basis to Itself is Identity

Theorem
Let $R$ be a ring with unity.

Let $M$ be an $n$-dimensional free $R$-module.

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

Then the change of basis matrix from $B$ to $B$ is the $n\times n$ identity matrix.

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