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Beginner question. From my understanding roughly, the Grassmannian $Gr(k, V)$ is a space which parametrizes all k-dimensional linear subspaces of the n-dimensional vector space $V$. A vector space is a set $V$ on which $+$ (addition) and $\ast$ (scalar multiplication) are defined. A linear subspace (or vector subspace) is just a subset of the vector space which is also a vector space. I'm just confused what it means to parameterize a space.

So my questions are:

  1. If a Grassmannian is just all subsets of dimension $k$ of the vector space $V$. That is, all possible combinations of subsets.
  2. If so, what it means to parameterize all those subsets. What the purpose is in parameterizing the vector space $V$. Wondering what/where the parameters are.
  3. What a Grassmanian looks like for Gr(2, 2). If that's not a thing then $Gr(2, 3)$ would be helpful too. Wikipedia says for $k=2$ it's the space of all planes through the origin. Wondering how they got there, I don't see the connection to "parameterizing all subsets of $V$". Not sure where the planes came from and such. Wondering what data is contained in its parameterized subsets.
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  • $\begingroup$ A linear subspace is not just "a subset of a vector space". It is a subset which is also a vector space. Therefore it has a basis (of cardinality $k$). Very roughly, $Gr(k,V)$ is the space of all such bases, up to the equivalence relation of "two bases span the same linear subspace". $\endgroup$
    – Michal Adamaszek
    May 8, 2018 at 13:12
  • $\begingroup$ For $k=2$ you get $2$-dimensional sub-sapces which can be seen as planes through the origin. Equally, $1$-dimensional subspaces can be seen as lines through the origin. $\endgroup$
    – M. Winter
    May 8, 2018 at 13:33
  • $\begingroup$ $\mathrm{Gr}(2,2)$ is a single point because there is only a single 2-dimensional subspace of $\Bbb R^2$, namely $\Bbb R^2$ itself. It holds that $\mathrm{Gr}(k,n)\cong \mathrm{Gr}(n-k,n)$, hence $\mathrm{Gr}(2,3)\cong\mathrm{Gr}(1,3)$ and hence $\mathrm{Gr}(2,3)$ is the projective space $P\Bbb R^3$, i.e. the set of all lines through the origin in $\Bbb R^3$. The first "interesting" one is $\mathrm{Gr}(2,4)$. $\endgroup$
    – M. Winter
    May 8, 2018 at 13:36

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To say that a space $X$ parametrizes some sort of things means that there's a one-to-one correspondence between the points in $X$ and the things of that sort. In your specific case, it means that there's a one-to-one correspondence between points in $Gr(k,V)$ and $k$-dimensional subspaces of $V$.

In principle, the $X$ in this situation could be merely a set, but often it's a topological space (or a smooth manifold, or an algebraic variety, or something of that sort). In that case, it includes a notion of "nearness" for the things it parametrized: two of those things are "close" if the corresponding points in $X$ are close. In the case of the Grassmannian, there's an intuitive idea of closeness between subspaces of a vector space. (For example, in the case of lines in $\mathbb R^3$, one can consider the angle between them; similarly for planes in $\mathbb R^3$. Higher dimensions get more complicated.) The topology on $Gr(k,V)$ matches this intuitive notion of closeness. (It turns out that Grassmannians are much better than mere topological spaces; they're smooth projective algebraic varieties, so in particular they're smooth compact manifolds.)

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