We give simple formulas for the canonical metric, gradient, Lie derivative, Riemannian connection, parallel translation, geodesics and distance on the Grassmann manifold of p-planes in R ⁿ . In these formulas, p-planes are represented as the column space of n×p matrices. The Newton method on abstract Riemannian manifolds proposed by Smith is made explicit on the Grassmann manifold. Two applications – computing an invariant subspace of a matrix and the mean of subspaces – are worked out.