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:
- If a Grassmannian is just all subsets of dimension $k$ of the vector space $V$. That is, all possible combinations of subsets.
- 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.
- 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.