In mathematics, the Grassmannian G r k ( V ) {\displaystyle \mathbf {Gr} {k}(V)} 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} . 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 earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to G r 2 ( R 4 ) {\displaystyle \mathbf {Gr} _{2}(\mathbf {R} ^{4})} , parameterizing them by what are now called Plücker coordinates. (See § Plücker coordinates and the Plücker relations below.) Hermann Grassmann later introduced the concept in general.

Notations for Grassmannians vary between authors, and include G r k ( V ) {\displaystyle \mathbf {Gr} {k}(V)} , G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} , G r k ( n ) {\displaystyle \mathbf {Gr} {k}(n)} , G r ( k , n ) {\displaystyle \mathbf {Gr} (k,n)} to denote the Grassmannian of k {\displaystyle k} -dimensional subspaces of an n {\displaystyle n} -dimensional vector space V {\displaystyle V} .