How a Grassmanian Parameterizes a Vector Space

Question

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.

Answer 1

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.)