Definition:Change of Basis Matrix

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

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

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

Let $I_G$ be the identity linear operator on $G$.

Let $\left[{I_G; \left \langle {a_n} \right \rangle, \left \langle {b_n} \right \rangle}\right]$ be the matrix of $I_G$ relative to $\left \langle {b_n} \right \rangle$ and $\left \langle {a_n} \right \rangle$.

Then $\left[{I_G; \left \langle {a_n} \right \rangle, \left \langle {b_n} \right \rangle}\right]$ is called the matrix corresponding to the change of basis from $\left \langle {a_n} \right \rangle$ to $\left \langle {b_n} \right \rangle$.

Note
Note the order of the above.

The original ordered basis is regarded as the ordered basis of the codomain of $I_n$, and the new ordered basis is regarded as the ordered basis of the domain of $I_n$.