Grassmannian

In mathematics, a Grassmannian G r k ( V ) {\displaystyle \mathbf {Gr} _{k}(V)} , also known as a Grassmann manifold, is a differentiable manifold that parameterizes the set of all k {\displaystyle k} -dimensional linear subspaces of an n {\displaystyle n} -dimensional vector space V {\displaystyle V} over a field K {\displaystyle K} that has a differentiable structure. For example, the Grassmannian G r 1 ( V ) {\displaystyle \mathbf {Gr} _{1}(V)} is the space of lines through the origin in V {\displaystyle V} , so it is the same as the projective space P ( V ) {\displaystyle \mathbf {P} (V)} of one dimension lower than V {\displaystyle V} . When V {\displaystyle V} is a real or complex vector space, Grassmannians are compact smooth manifolds, of dimension k ( n − k ) {\displaystyle k(n-k)} . In general they have the structure of a nonsingular projective algebraic variety. The Grassmannian is named for the German polymath, linguist and mathematician Hermann Grassmann, who introduced the concept to mathematics.