Rank & Nullity

A matrix A takes an input vector and produces an output. Some outputs are reachable — that set is the column space, and its dimension is the rank. Some inputs get flattened to zero — that set is the null space, and its dimension is the nullity. The stunning fact: these two dimensions always add up to the number of input columns. The rank-nullity theorem is a conservation law — what A reaches plus what it kills exactly equals what was fed in.