The Grassmannian G(k,n), named after Hermann Grassmann (1809–77), is a smooth manifold parameterizing all the k-dimensional subspaces of ℝⁿ. When k = 1, this is (n–1)-dimensional real projective space. When k = 2, a subspace can be determined by a basis v₁,v₂. Unfortunately, there are many different bases for the same subspace. However, if ∧ denotes the exterior product, then

and so any two bases give the same product in ⋀²ℝⁿ up to a scalar multiple. This means that G(2,n) can be considered as a submanifold of the projective space ℙ(⋀²ℝⁿ) and more generally G(k,n) ⊆ ℙ(⋀^kℝⁿ). The dimension of G(k,n) equals k(n−k).