In mathematics, the Grassmannian Gr(k, V) is a space that parameterizes all k-dimensional linear subspaces of the n-dimensional vector space V. For example, the Grassmannian Gr(1, V) is the space of lines through the origin in V, so it is the same as the projective space of one dimension lower than V.

When V is a real or complex vector space, Grassmannians are compact smooth manifolds. In general they have the structure of a smooth algebraic variety, of dimension ${\displaystyle k(n-k).}$

The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in projective 3-space, equivalent to Gr(2, R⁴) and parameterized them by what are now called Plücker coordinates. Hermann Grassmann later introduced the concept in general.

Notations for the Grassmannian vary between authors; notations include Gr_k(V), Gr(k, V), Gr_k(n), or Gr(k, n) to denote the Grassmannian of k-dimensional subspaces of an n-dimensional vector space V.