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

Throughout we use the axioms of a vector space over $K$ and the two conditions of linearity, which $S$ and $T$ satisfy because they are linear operators on $V$. Let $u,v\in V$ and $\mu\in K$.

Claim 1. Write $A$ for the map sending $u$ to $S(u)+T(u)$. Using the linearity of $S$ and $T$ and then the commutativity and associativity of vector addition,

Using linearity again and the vector space identity $\mu(x+y)=\mu x+\mu y$,

Hence $A$ is linear and maps $V$ to $V$, so it is a linear operator on $V$.

Claim 2. Write $B$ for the map sending $u$ to $\lambda S(u)$. Then

using linearity of $S$ and $\lambda(x+y)=\lambda x+\lambda y$. Also, using linearity of $S$, the vector space identity $(\alpha\beta)x=\alpha(\beta x)$ twice, and the commutativity of multiplication in the field $K$,

Hence $B$ is a linear operator on $V$.

Claim 3. Write $C$ for the map sending $u$ to $S(T(u))$. By the linearity of $T$ and then of $S$,

and

Hence $C$ is a linear operator on $V$.

Claim 4. Write $I$ for the map sending $u$ to $u$. Then $I(u+v)=u+v=I(u)+I(v)$ and $I(\mu u)=\mu u=\mu\,I(u)$, so $I$ is a linear operator on $V$.