Definition:Change of Basis Matrix/Definition 1

Definition
Let $R$ be a commutative ring with unity.

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

Let $A=\left \langle {a_n} \right \rangle$ and $B=\left \langle {b_n} \right \rangle$ be ordered bases of $G$.

The matrix of change of basis from $A$ to $B$ is the matrix whose columns are the coordinate vectors of the elements of the new basis $\left \langle {b_n} \right \rangle$ relative to the original basis $\left \langle {a_n} \right \rangle$.