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 $$I_n$$.