Grassmannian

In mathematics, a Grassmannian ${\displaystyle \mathbf {Gr} _{k}(V)}$, also known as a Grassmann manifold, is a differentiable manifold that parameterizes the set of all ${\displaystyle k}$-dimensional linear subspaces of an ${\displaystyle n}$-dimensional vector space ${\displaystyle V}$ over a field ${\displaystyle K}$ that has a differentiable structure. For example, the Grassmannian ${\displaystyle \mathbf {Gr} _{1}(V)}$ is the space of lines through the origin in ${\displaystyle V}$, so it is the same as the projective space ${\displaystyle \mathbf {P} (V)}$ of one dimension lower than ${\displaystyle V}$. When ${\displaystyle V}$ is a real or complex vector space, Grassmannians are compact smooth manifolds, of dimension ${\displaystyle k(n-k)}$. 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.