rank of the transpose
Here is one of the most quietly astonishing facts in a first course: a matrix and its transpose have the same rank. Equivalently, the dimension of the column space equals the dimension of the row space — row rank equals column rank. From the entries alone there is no obvious reason rows and columns should agree, yet they always do.
Duality explains why with almost no computation. For a linear map T : V -> W, the transpose T* : W* -> V* satisfies rank T = rank T*. Reason: the image of T* is the annihilator of the kernel of T (up to the canonical identifications), and dimension counting via dim U + dim U^0 = dim V then forces the two ranks to match.
Reading it through matrices: rank T is the column rank of A (dimension of the column space), and rank T* is the column rank of A^T, which is the row rank of A. So 'rank T = rank T*' is literally 'column rank = row rank' once you translate. The mysterious symmetry of rows and columns is just the symmetry of a map and its dual.
This is the payoff of taking duality seriously: a theorem that looks like a computational coincidence becomes a one-line structural consequence. It also generalizes cleanly to infinite dimensions and to maps that have no convenient matrix, where elementary row-reduction arguments simply do not apply.
Row rank equals column rank because a map and its dual have equal rank.
Chain of reasoning: im(T*) = (ker T)^0, so rank T* = dim V - dim ker T = rank T by rank-nullity. Three duality facts collapse into the single most-used equality of elementary linear algebra.