Definition:Change of Basis Matrix/Definition 2

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

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

Let $A = \sequence {a_n}$ and $B = \sequence {b_n}$ be ordered bases of $G$. Let $I_G$ be the identity linear operator on $G$.

Let $\sqbrk {I_G; \sequence {a_n}, \sequence {b_n} }$ be the matrix of $I_G$ relative to $\sequence {b_n}$ and $\sequence {a_n}$.

Then $\sqbrk {I_G; \sequence {a_n}, \sequence {b_n} }$ is called the matrix corresponding to the change of basis from $\sequence {a_n}$ to $\sequence {b_n}$.