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I am studying some material related to Grassmannians and in particular how to representk-subspaces of ℝ^{n}as "points" in another space.

I think understood the general idea behind the Plücker embedding, however, I recently came across another type of embedding (the "Projection embedding") that sounds more intuitive and simpler to understand (see attached figure for its definition).

Can anyone elaborate a bit more on the main differences between Plücker and Projection embeddings?

In the past I browsed some old textbooks in the classical literature of algebraic geometry, and while the Plücker embedding is always treated extensively, the Projection embedding is not even mentioned at all. Why?

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# A Representation of elements of the Grassmannian space

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