Operations on Linear Operators

Let $K$ be a field and let $V$ be a vector space over $K$. The set of linear operators on $V$ carries the following operations, each of which again yields a linear operator on $V$ by Sums, Scalar Multiples, Composites and the Identity are Linear Operators. Let $S$ and $T$ be linear operators on $V$ and let $\lambda\in K$.

The sum $S+T$ is the map sending $u\in V$ to $S(u)+T(u)$.

The scalar multiple $\lambda S$ is the map sending $u\in V$ to $\lambda\,S(u)$.

The product $ST$ is the map sending $u\in V$ to $S(T(u))$.

The identity operator $\mathrm{id}_{V}$ is the map sending $u\in V$ to $u$.