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 $\BB$ be an ordered basis of $M$.

Then the change of basis matrix from $\BB$ to $\BB$ is the $n\times n$ identity matrix:

$\mathbf M_{\BB, \BB} = \mathbf I$

Proof

Follows directly from the definition of change of basis matrix.

$\blacksquare$