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In mathematics, the Grassmannian $\mathbf{Gr}(r, V)$ is a space which parameterizes all linear subspaces of a vector space $V$ of given dimension $r$. For example, the Grassmannian $\mathbf{Gr}(1, V)$ 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 (Wikipedia).

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