rank-nullity (abstract form)

Volume I proved rank-nullity by counting pivot and free columns in a row-reduced matrix. The abstract form proves the same equation, dim ker T + dim im T = dim V, with no matrix at all — and in doing so reveals why it is true rather than merely that it is true.

Start from the first isomorphism theorem: V / ker T is isomorphic to im T. Isomorphic spaces have equal dimension, so dim(V/ker T) = dim im T. Now use the dimension formula for quotients, dim(V/U) = dim V - dim U, applied with U = ker T. That gives dim V - dim ker T = dim im T, which rearranges to exactly dim ker T + dim im T = dim V. The whole proof is two facts chained together.

Reading the equation: the rank dim im T counts the directions T genuinely outputs, and the nullity dim ker T counts the directions T destroys. Every dimension of the domain is accounted for as either preserved (contributing to rank) or collapsed (contributing to nullity). Nothing is created or lost — dimension is conserved across the map.

This abstract proof is more than elegance. It works verbatim for maps between infinite-dimensional spaces wherever the quotient still has finite dimension, it survives any change of basis because no basis was chosen, and it is the template for the rank theorems of modules and chain complexes. The matrix proof computes; the abstract proof explains.