Loterre: Mathematics: Grassmannian

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Grassmannian

In mathematics, the Grassmannian ${\\displaystyle \\mathbf {Gr} _{k}(V)}$ 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 ${\\displaystyle K}$ .
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.
(Wikipedia, The Free Encyclopedia, https://en.wikipedia.org/wiki/Grassmannian)