TITLE: linear transformation $T$ such that $TS = ST$
QUESTION [8 upvotes]: Let $V$ be a finite-dimensional vector space over $F$. Let $T:V \rightarrow V$ be a linear transformation such that $ST=TS$ for all linear transformations $S:V \rightarrow V$. Show that $T = cI_v$ for some $c \in F$.

REPLY [9 votes]: Notice that if two linear operators $A,B$ commute, they preserve one another's eigenspaces. It remains to show that a transformation which preserves every subspace (so in particular, every vector is its eigenvector) is a multiple of identity.
Note that for this argument, you don't need to assume that $V$ is finite-dimensional.