# 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 $G$.

### Definition 1

The matrix of change of basis from $A$ to $B$ is the matrix whose columns are the coordinate vectors of the elements of the new basis $\left \langle {b_n} \right \rangle$ relative to the original basis $\left \langle {a_n} \right \rangle$.

### Definition 2

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, 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 $I_n$.