transpose of a linear map
Every linear map T : V -> W induces a map running the other way between the duals, called its transpose (or dual map) T* : W* -> V*. The recipe is precomposition: given a functional g on W, you produce a functional on V by first applying T and then g. In symbols, T*(g) = g composed with T, so (T*g)(v) = g(T v).
Read that arrow direction carefully: T goes V -> W, but T* goes W* -> V*. Dualizing flips the arrow. This is not a quirk of bookkeeping; it is what 'measuring the output' means. If g measures things in W, then g-after-T measures things in V, namely 'apply T, then measure'.
When you write everything in dual bases, the matrix of T* is exactly the transpose A^T of the matrix A of T. So the humble operation of flipping a matrix across its diagonal is revealed as a coordinate-free, basis-independent construction. The transpose was a real map all along; the flipped matrix is just its shadow.
Transpose plays well with composition and identity: (S T)* = T* S* (note the reversal) and the transpose of the identity is the identity. These functorial rules let you dualize entire diagrams of maps at once, which is how the four-subspace duality and rank equalities fall out.
Transpose precomposes a functional with T; in dual bases this is matrix transpose.
Watch the order reversal: (S T)* = T* S*. Putting a map in dual coordinates and then transposing the matrix gives the same answer as transposing the map abstractly — that compatibility is exactly why A^T is the 'right' definition of matrix transpose.