I have recently learnt about group actions and grassmannian space. So While searching further on the topic about general linear groups acting on different sets, I read something on wikipedia:
$GL$($V$) acts transitively on grassmannian space
But I didn't find any prove regarding this statement.
My attempt was to define a map that takes the basis from grassmannian set to another basis within the set, but I couldn't really fill out the details and I am not sure how to construct a prove for this statement.
For each $T \in {\rm GL}(V)$, you can define $\widetilde{T}:{\rm Gr}_k(V) \to {\rm Gr}_k(V)$ by $\widetilde{T}(W) = T[W]$, where the latter denotes a direct image. This construction has a functorial behavior, that is:
$\widetilde{T\circ S} = \widetilde{T}\circ \widetilde{S}$.
$\widetilde{{\rm Id}_V} = {\rm Id}_{{\rm Gr}_k(V)}$.
$\widetilde{T^{-1}} = \widetilde{T}^{-1}$.
This means we have an action ${\rm GL}(V) \circlearrowright {\rm Gr}_k(V)$ given by evaluation preceded with taking tilde. Given two $W,W'\in {\rm Gr}_k(V)$ take bases for $W$ and $W'$, complete them to bases for $V$, and define $T$ by mapping the basis for $W$ onto the basis for $W'$, and the remainder of one completed basis to the remainder of the other. Then $\widetilde{T}(W)=W'$ by construction.