# 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 $I_n$.

## 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
- Results about
**change of basis**can be found here.