Bump

Given a $n$-dim vector space $V$ with ground field $K$, the set of ordered basis of $V$ is a $GL_n(K)$-torsor

$GL_n(K)$ acts on them freely and transitively

it is a "group without identity"

If we pick a basis $B=(v_1, v_2, v_3, \cdots, v_n)$ then we can make it a group: if $C$ and $D$ are basis then define $C \cdot D$ to be $E$ such that $[id]_{EB} = [id]_{DB} [id]_{CB}$