I have come across a definition of rank in some lecture notes I am using to prepare for a linear algebra qualifying exam and the notes define the rank of a linear map in a somewhat different manner than I am used to.
let $V$ be a finite dimensional vector space and $f \in End(V)$. Since $End(V) \cong V^* \otimes V$ we can provide a new definition of the rank of $f$.
How do we show $dim(Im(f)) = min \{ t : f = \sum_{i=1}^{t} v_{i}^{*} \otimes v_i \}$ where $v_{i}^{*} \in V^*$ and $ v_i \in V$?