Sums, Scalar Multiples, Composites and the Identity are Linear Operators

Let $K$ be a \reftext{def:field-c54-2026b}{field}, let $V$ be a \reftext{def:vector-space-2026a}{vector space over $K$}, let $S$ and $T$ be \reftext{def:linear-operator-2026a}{linear operators} on $V$, and let $\lambda\in K$. Then each of the following maps from $V$ to $V$ is a linear operator on $V$.

\textbf{1. (Sum)} The map sending $u\in V$ to $S(u)+T(u)$.

\textbf{2. (Scalar multiple)} The map sending $u\in V$ to $\lambda\,S(u)$.

\textbf{3. (Composite)} The map sending $u\in V$ to $S(T(u))$.

\textbf{4. (Identity)} The map sending $u\in V$ to $u$.