# 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 $\AA$ and $\BB$ be ordered bases of $M$.

Let $\mathbf P$ be the change of basis matrix from $\AA$ to $\BB$.

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

## Proof

$\sqbrk {I_M; \AA, \BB} \sqbrk {I_M; \BB, \AA} = \sqbrk {I_M; \AA, \AA} = I_n$
$\sqbrk {I_M; \BB, \AA} \sqbrk {I_M; \AA, \BB} = \sqbrk {I_M; \BB, \BB} = I_n$

Hence the result.

$\blacksquare$