Grassmannian - Duality

Duality

Every r-dimensional subspace W of V determines an (n–r)-dimensional quotient space V/W of V. This gives the natural short exact sequence:

Taking the dual to each of these three spaces and linear transformations yields an inclusion of (V/W)* in V* with quotient W*:

Using the natural isomorphism of a finite-dimensional vector space with its double dual shows that taking the dual again recovers the original short exact sequence. Consequently there is a one-to-one correspondence between r-dimensional subspaces of V and (n–r)-dimensional subspaces of V*. In terms of the Grassmannian, this is a canonical isomorphism

Gr(r, V) ≅ Gr(n − r, V*).

Choosing an isomorphism of V with V* therefore determines a (non-canonical) isomorphism of Gr(r, V) and Gr(n−r, V). An isomorphism of V with V* is equivalent to a choice of an inner product, and with respect to the chosen inner product, this isomorphism of Grassmannians sends an r-dimensional subspace into its (n–r)-dimensional orthogonal complement.