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In mathematics, a Grassmannian , also known as a Grassmann manifold is a differentiable manifold that parameterizes the set of all -dimensional linear subspaces of an -dimensional vector space over a field that has a differentiable structure. For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective space of one dimension lower than .
When is a real or complex vector space, Grassmannians are compact smooth manifolds, of dimension . 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.
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