Learn Use oriented incidence matrices to connect conservation, cycles, potentials and orthogonal certificates. Signed real flows are allowed; capacities and nonnegative-flow constraints require additional models.
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Network circulations live in edge coordinates, while conservation demands live in vertex coordinates.
Rank-nullity gives nullity six minus four, or two. The column count is the domain dimension; it must not be replaced by the number of vertex rows.
The minimum-flow proof compares a feasible flow with every circulation, not just a few plotted alternatives.
If f is perpendicular to every allowed displacement z, then ||f+z|| squared equals ||f|| squared plus ||z|| squared. Every nonzero displacement increases the squared norm.
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Construct bases in vertex and edge coordinates, then prove how connected components and cycles determine the four spaces.
6 practice tasks.
Use component balances to certify feasibility, then select a unique minimum-norm signed flow through vertex potentials.
8 practice tasks.