Take two linear maps T \in \mathcal{L}(U,V) and S \in \mathcal{L}(V,W), then ST \in \mathcal{L}(U,W) is defined by:
\begin{equation} (ST)(u) = S(Tu) \end{equation}
Indeed the “product” of Linear Maps is just function composition. Of course, ST is defined only when T maps to something in the domain of S. The following there properties hold on linear-map products (note that commutativity isn’t one of them!): associativity \begin{equation} (T_1T_2)T_3 = T_1(T_2T_3) \end{equation} identity \begin{equation} TI = IT = T \end{equation} for T \in \mathcal{L}(V,W) and I \in \mathcal{L}(V,V) (OR I \in \mathcal{L}(W,W) depending on the order) is the identity map in V. identity commutes, as always. distributive in both directions—
\begin{equation} (S_1+S_2)T = S_1T + S_2T \end{equation}
and
\begin{equation} S(T_1+T_2) = ST_{1}+ST_{2} \end{equation}