Let $R$ be a commutative ring with identity, and let $m,n \in \mathbb{Z}$ with $m,n \geq 0$. Let $R^n$ and $R^m$ denote the finite free $R$-modules with standard bases $(e_1,\dots,e_n)$ and $(f_1,\dots,f_m)$, respectively, with the convention that $R^0$ has the empty standard basis.

For every $R$-[module homomorphism](/page/Module%20Homomorphism) $\varphi: R^n \to R^m$, define its standard matrix $A_\varphi = (a_{ij}) \in R^{m \times n}$ by the identities

\begin{align*} \varphi(e_j)=\sum_{i=1}^{m} a_{ij} f_i \end{align*}

for every $j \in \{1,\dots,n\}$. Then the assignment

\begin{align*} \operatorname{Hom}_R(R^n,R^m) &\to R^{m \times n} \end{align*}

\begin{align*} \varphi &\mapsto A_\varphi \end{align*}

is a bijection.

Moreover, if $p \in \mathbb{Z}$ with $p \geq 0$, if $\varphi: R^n \to R^m$ has standard matrix $A \in R^{m \times n}$, and if $\psi: R^m \to R^p$ has standard matrix $B \in R^{p \times m}$, then $\psi \circ \varphi: R^n \to R^p$ has standard matrix $BA \in R^{p \times n}$, where matrices act on column vectors by ordinary matrix multiplication.