We learned in class that the number of independent KCL equations is N-1(N-number of nodes) and the number of independent KVL equations is B-(N-1) (B- number of branches). Thus we have B independent equations and we can solve the circuit for the currents. What I don't understand is how can we know for sure that the B equations are independent? The union of two independent sets of vectors is not necessarily also independent.
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The equations obtained from the KCL involve only currents; those obtained from the KVL only voltages. Their union cannot thus form a linear dependent set of equations, if the two sets are singularly independent.
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\$\begingroup\$ The equations obtained from KVL also involve current as you are writing each v as v=IR \$\endgroup\$– VenuceCommented Oct 26, 2018 at 13:56
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\$\begingroup\$ @Venuce Those are additional B relationships which are the descriptive equations of the elements, but equations obtained from KVL involve only voltages. There are 2B unkowns overall, and thus you need 2B equations: N-1 from the KCL, B-(N-1) from KVL and B descriptive equations, one for each branch. \$\endgroup\$ Commented Oct 26, 2018 at 14:02