I am trying to prove the following theorem: Let $G$ be a graph. The cycle space and cut space are orthogonal complement if and only if the graph $G$ has an odd number of spanning tree.
My attempt:
I define the matrix $H=\left [ \begin{array}{c|c} Q \\ B\end{array} \right ] $ where $Q$ and $B$ are cut matrix and circuit matrix respectively. I thought about $|H^tH|$.
