For a directed graph $(V,E),$ there is a linear map $\partial:\Bbb R[E]\rightarrow\Bbb R[V]$ defined by $\partial (a,b)=b-a.$ Prove that the following are equivalent for a directed graph $(V, E).$ $(1)\; \#V−\#E= 1.\\(2)\; (V, E) \text{ is a tree, i.e. } (V\text{, } E) \text{ has no non-trivial cycles and is connected.}$
To prove this, I need to use the fact that $(V\text{, }E)\text{ has no cycles iff ker } \partial= 0.$
My general idea is that if $(V,E)$ has no non-trivial cycles, then $\partial$ is injective. This implies that $\#E=\text{dim im }\partial.$ Furthermore, connectedness $\implies \#V-\text{dim im }\partial=1\implies \#V-\#E=1$