kernel-image duality

Dualizing a linear map swaps kernels and images in a precise, beautiful pattern. For T : V -> W with transpose T* : W* -> V*, the kernel of T* is exactly the annihilator of the image of T: ker(T*) = (im T)^0. A functional g on W is killed by T* precisely when g vanishes on everything T can reach.

There is a companion identity going the other way: im(T*) = (ker T)^0, the image of the transpose is the annihilator of the kernel of T. Together these two equations are the duality form of the 'four fundamental subspaces' you met with matrices — column space, null space, row space, left null space — now seen as annihilators of one another.

These follow from nothing but the definition of the pairing. g is in ker T* means T* g = 0, means <T* g, v> = 0 for all v, means <g, T v> = 0 for all v, means g vanishes on im T — which is the statement g is in (im T)^0. The proof is a single unwinding of <g, T v> = <T* g, v>; no coordinates, no row reduction.

The payoff is conceptual unity. Solvability of T x = b (is b in im T?) becomes a question about functionals (does every g in ker T* satisfy g(b) = 0?). This 'Fredholm alternative' flavor — solvable exactly when obvious obstructions vanish — is one of the most reusable ideas duality offers, and it survives intact into infinite dimensions.