Linear Transformations and Identity Matrix

Question

Suppose that $V$ is a finite dimensional vector space and $T$ is a linear transformation from $V$ to $V$. Prove that $T$ is a scalar multiple of the identity matrix iff $ST=TS$ for every linear transformation $S$ from $V$ to $V$.

Answer 1

One direction is trivial. For the other, take $\{v_1,...,v_n\}$ to be any basis for $V$, and consider the transformations $E_{i,j}:V\to V$ induced by $$v_k\mapsto\begin{cases} v_j & \text{if }k=i,\\0 & \text{otherwise.}\end{cases}$$ What can we conclude about $T$ from the fact that $E_{i,j}T=TE_{i,j}$ for all $i,j\in\{1,...,n\}$?