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 $\BB$ and $\CC$ be bases of $M$.

Let $\mathbf M_{\BB, \CC}$ be the change of basis matrix from $\BB$ to $\CC$.

Let $m \in M$.

Let $\sqbrk m_\BB$ and $\sqbrk m_\CC$ be its coordinate vectors relative to $\BB$ and $\CC$ respectively.

Then $\sqbrk m_\BB = \mathbf M_{\BB, \CC} \cdot \sqbrk m_\CC$.

Proof
Intuitively, when we consider $\BB$ and $\CC$ as row vectors, this is because:
 * $\CC = \BB \cdot \mathbf M_{\BB, \CC}$ and:
 * $\BB \cdot \sqbrk m_\BB = \CC \cdot \sqbrk m_\CC$ imply:
 * $\BB \cdot \sqbrk m_\BB = \BB \cdot \mathbf M_{\BB, \CC} \cdot \sqbrk m_\CC$.

Because $\BB$ is a basis, this implies $\sqbrk m_\BB = \mathbf M_{\BB, \CC} \cdot \sqbrk m_\CC$.

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