Grassmannian - The Grassmannian As A Set

The Grassmannian As A Set

Let V be a finite-dimensional vector space over a field k. The Grassmannian Gr(r, V) is the set of all r-dimensional linear subspaces of V. If V has dimension n, then the Grassmannian is also denoted Gr(r, n).

Vector subspaces of V are equivalent to linear subspaces of the projective space P(V), so it is equivalent to think of the Grassmannian as the set of all linear subspaces of P(V). When the Grassmannian is thought of this way, it is often written as Gr(r' −1, P(V)) or Gr(r−1, n−1).