I want to solve this problem: A k-frame in $R_{n}$ is a k-tuple of vectors in $R_{n} (f_{1},..., f_{k})$ that are orthonormal. Let $V_{k,n}$ denote the set of all k-frames in $R_{n}$. Show that $V_{k,n}$ is a homogeneous space.
$\textbf{My attempt:}$
Orthonormality implies $f_{i}f_{j}=\delta_{ij}$.
If $V_{k,n}$ is a homogeneous space, it should be written as a coset G/H with $V_{k,n}\equiv G$. Thus, we must find an equivalence relation $g'\sim g \quad \text{iff} \quad \exists h\in H \quad \text{such that} \quad g'=gh$.
We can write $(f_{1},f_{2},...,f_{k})=f_{a}^{-1}(f_{1}f_{a}+f_{2}f_{a}+...+f_{a}f_{a}+...+f_{k}f_{a})=f_{a}^{-1}(\delta_{1a}+\delta_{2a}+...+0+...+\delta_{ka})$
Thus, the subgroup H could be a set of all tuples $H={f_{1},...,f_{k}}$ of a frame and there is an $h=f_{a}$ of H such that $g'=gh$ where $g'=(\delta_{1a}+\delta_{2a}+...+0+...+\delta_{ka})$ and $g=(f_{1},f_{2},...,f_{k})$
Does this make sense? What do you think?