Suppose $S,T\in \mathcal{L}(V,W)$ and $\lambda \in F$. The sum $S+T$ and product $\lambda T$ are the linear maps from $V$ to $W$ defined by
$
(S+T)(v)=Sv+Tv
$
and
$
(\lambda T)(v)=\lambda(Tv)
$
Additionally, if $T\in \mathcal{L}(U,V)\text{ and }S\in \mathcal{L}(V,W)$, then the product $ST\in \mathcal{L}(U,W)$ is defined by
$
(ST)(u)=S(Tu)
$
for $u\in U$.
Linear maps are associative.
$
(T_{1}T_{2})T_{3}=T_{1}(T_{2}T_{3})
$
They are also distributive.
$
\begin{align}
(S_{1}+S_{2})T&=S_{1}T+S_{2}T\\
&\text{and}\\
S(T_{1}+T_{2})&=ST_{1} +ST_{2}
\end{align}
$