Let $V$ be a finite-dimensional space, and let ${\cal L}(V)$ denote the space of all endomorphisms of $V$.
For any $\phi \in {\cal L}({\cal L}(V))$, there is a unique bilinear map ${\cal L}(V) \otimes {\cal L}(V) \to {\cal L}(V)$ (which we denote by ${\Phi}(\phi)$), satisfying
$ \Phi(\phi) (a \otimes b)=\phi(ab) $
for any $a,b\in {\cal L}(V)$. The $\Phi$ thus defined is obviously linear in $\phi$. What is its rank ?