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}