Exercise 3.D.17

Proof. Note that if $I\in E$, then $E=L(V)$. We will prove $I\in E$ if $E$ is not $0$.

Suppose $E\ne 0$. Then there exists non-zero $T\in E$. Let $v_1\in V$ such that $T{v}_1\ne 0$. Then $v_1$ is also non-zero and we can extend it to a basis $v_1,\dots ,v_n$ of $V$. Define $S_1,\dots ,S_n\in L(V)$ by

and let $R_1,\dots ,R_n$ be any operators on $V$ such that

for each $j=1,\dots ,n$. Note that $R_{j}T{S}_j\in E$. We have

and, for $k\ne j$,

Thus $R_{1}T{S}_1+\cdots +R_{n}T{S}_n$ equals the identity and is in $E$, since $E$ is closed under addition. □