Tangent Spaces to Grassmannians

We have seen that the Grassmannian ??(k, n) is a smooth variety of dimension (k + 1) (n - k). This follows initially from our explicit description of the covering of ?? (k, n) by open sets U_Λ ≅ ??(k+1)(n-k), though we could also deduce this from the fact that it is a homogeneous space for the algebraic group PGL _n+1K. The Zariski tangent spaces to G are thus all vector spaces of this dimension. For many reasons, however, it is important to have a more intrinsic description of the space T_Λ(??;) in terms of the linear algebra of Λ ⊂ Kn+1. We will derive such an expression here and then use it to describe the tangent spaces of the various varieties constructed in Part I with the use of the Grassmannians.