What is the rank of a linear transformation space? $L(V,W)$

Question

What does the $\operatorname{rank}(f(T))$ mean exactly, if $f:L(V,W) \to L(V,W)$ and $T \in L(V,W)$, where $L(V,W)$ is the set of all linear transformations between two finite dimensional vector spaces $V$ and $W$?

Answer 1

Well, $f$ is a map assigning to each linear map $T$ a linear map $f(T)$. Therefore, $\operatorname{rank}(f(T))$ is the rank of said linear map $f(T)$: $$\operatorname{rank}(f(T))=\dim\left( f(T)(V)\right)$$