If an arrow map failed to have maximal <matrix rank>, it would have both a nonzero <kernel of a linear map> and a nonzero <cokernel>. Apply the preceding obstruction with $Y=X$. It would give $\operatorname{Ext}^1_Q(X,X)\ne0$, contrary to the assumption. Therefore \b[every arrow map is injective or surjective], equivalently its rank is the minimum of the two vertex dimensions. This is a necessary property of a <rigid quiver representation>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 3 / 3 / b / i / Solution

 Solution (source code)