Definition:Change of Basis Matrix/Warning

Definition
Let $R$ be a ring with unity.

Let $G$ be a finite-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 corresponding to the change of basis from $\sequence {a_n}$ to $\sequence {b_n}$.

Note the order of the above, which gives rise to the intuitive cancellation law in Product of Change of Basis Matrices.

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 $I_n$.