Operations on Linear Operators

Let $K$ be a \reftext{def:field-c54-2026b}{field} and let $V$ be a \reftext{def:vector-space-2026a}{vector space over $K$}. The set of \reftext{def:linear-operator-2026a}{linear operators} on $V$ carries the following operations, each of which again yields a linear operator on $V$ by \ref{lem:operator-operations-linear-2026a}. Let $S$ and $T$ be linear operators on $V$ and let $\lambda\in K$.

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

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

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

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