Relative Matrix of Composition of Linear Transformations

Theorem
Let $R$ be a ring with unity.

Let $M,N,P$ be free $R$-modules of finite dimension $m,n,p>0$ respectively.

Let $\mathcal A,\mathcal B,\mathcal C$ be ordered bases of $M,N,P$.

Let $f:M\to N$ and $g:N\to P$ be linear mappings, and $g\circ f$ be their composition.

Let $\mathbf M_{f, \mathcal B, \mathcal A}$ and $\mathbf M_{g, \mathcal C, \mathcal B}$ be their matrices relative to $\mathcal A,\mathcal B$ and $\mathcal B,\mathcal C$ respectively.

Then the matrix of $g\circ f$ relative to $\mathcal A$ and $\mathcal C$ is:
 * $\mathbf M_{g\mathop\circ f, \mathcal C, \mathcal A} = \mathbf M_{g, \mathcal C, \mathcal B}\cdot \mathbf M_{f, \mathcal B, \mathcal A}$

Proof
Let $\mathcal A = \left \langle {a_m} \right \rangle$, $\mathcal B = \left \langle {b_n} \right \rangle$, $\mathcal C = \left \langle {c_p} \right \rangle$.

Let $\left[{\alpha}\right]_{m n} = \left[{f; \left \langle {b_n} \right \rangle, \left \langle {a_m} \right \rangle}\right]$ and $\left[{\beta}\right]_{n p} = \left[{g; \left \langle {c_p} \right \rangle, \left \langle {b_n} \right \rangle}\right]$.

Then:

Also see

 * Composition of Linear Mappings is Linear Mapping