Change of Coordinate Vector Under Change of Basis

Theorem
Let $R$ be a ring with unity.

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

Let $\mathcal B$ and $\mathcal C$ be bases of $M$.

Let $\mathbf M_{\mathcal B, \mathcal C}$ be the change of basis matrix from $\mathcal B$ to $\mathcal C$.

Let $m\in M$.

Let $[m]_{\mathcal B}$ and $[m]_{\mathcal C}$ be its coordinate vectors relative to $\mathcal B$ and $\mathcal C$ respectively.

Then $[m]_{\mathcal B} = \mathbf M_{\mathcal B, \mathcal C} \cdot [m]_{\mathcal C}$.

Proof
Intuitively, when we consider $\mathcal B$ and $\mathcal C$ as row vectors, this is because:
 * $\mathcal C = \mathcal B \cdot \mathbf M_{\mathcal B, \mathcal C}$ and:
 * $\mathcal B \cdot [m]_{\mathcal B} = \mathcal C \cdot [m]_{\mathcal C}$ imply:
 * $\mathcal B \cdot [m]_{\mathcal B} = \mathcal B \cdot \mathbf M_{\mathcal B, \mathcal C} \cdot [m]_{\mathcal C}$.

Because $\mathcal B$ is a basis, this implies $[m]_{\mathcal B} = \mathbf M_{\mathcal B, \mathcal C} \cdot [m]_{\mathcal C}$.

This can be formalized by giving $R\times M$ the structure of a ring. Alternatively, this can be verified directly, which boils down to re-proving that that matrix multiplication is associative.

Also see

 * Change of Coordinate Vectors Under Linear Mapping, an analogous result for linear transformations, of which this is a special case
 * Matrix Corresponding to Change of Basis under Linear Transformation