The book Graphs and Matrices by Bapat formulates linear algebra on graph theory, yet I cannot find important theorems such as Duality theorem between the cycle space and the cut space (Diestel p.26, Graph Theory): the cycle space and the cut space of any graph satisfy that the cycle space is the orthogonal complement of the cut space and vice versa. The Bapat book mentions on the page 64
"Let $Q$ be the [vertex-edge] incidence matrix of [graph] $G$. There is a close relationship between $Q$,[fundamental cut matrix] $B$ and [fundamental cycle matrix] $C$, as we see next."
after which Theorem 5.6 in which a cut matrix $B_f$ formulated in terms of the reduced incidence matrix and a cycle matrix $C_f$ similarly
"Let $Q_1$ be the reduced incidence matrix obtained by deleting the last row of $Q$ and suppose $Q_1$ is partitioned as $Q_1=[Q_{11},Q_{12}]$, where $Q_{11}$ is of order $(n−1)×(n−1)$.Then $B_f= Q^{−1}_{11} Q_{12}$ and $C_f=−Q_{12} (Q_{11})^{−1}$."
from which is it possible to deduce the duality theorem? In other words
How to formulate the duality theorem between the cycle space and the cut space of any graph in terms of linear algebra and matrices?
