Definition:Change of Basis Matrix

Definition
Let $R$ be a ring with unity.

Let $G$ be a finite-dimensional free $R$-module.

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

Note
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 $I_n$.

Also see

 * Equivalence of Definitions of Change of Basis Matrix
 * Change of Basis Matrix is Invertible
 * Change of Coordinate Vector Under Change of Basis
 * Bases of Free Module have Equal Cardinality, which means that the change of basis matrix is a square matrix
 * Bases of Vector Space have Equal Cardinality