> [!NOTE] Corollary (Composing Linear Maps) > Let $\psi:U\to V$ and $\varphi:V\to W$ be linear maps between FDVS. > Let $B$ be a basis of $U$, $B^{\prime}$ a basis of $V$, and $B^{\prime\prime}$ a basis of $W$ of sizes $\ell,n,m$ respectively. > > Suppose $\psi$ is represented by a matrix $A\in\mathrm{Mat}_{n\ell}$ with respect to these bases and $\varphi$ is represented by $A^{\prime}\in\mathrm{Mat}_{mn}$. Then there is a commutative square: $\begin{CD} U @> \varphi \, \circ \, \psi >> W\\ @VV\chi_{B} V @VV \chi_{B''} V\\ \mathbb{R}^{l} @>{ L_{A'} \, \circ \, L_{A} = L_{A'A} }>> \mathbb{R}^{m} \end{CD}$That is, the composition $\varphi \circ \psi$ is represented by the product $A'A\in \text{Mat}_{ml}$ wrt the bases of $U$ and $W$. > *Proof*. Follows from [[Commutative Square#^812a94|commutative squares can be glued together]].