Is it possible to find a set $V=\{V_1,...,V_n\}$ of $m$ dimensional distinct subspaces $V_i \subset \mathbb{R}^n$ so that for any linearly independent subset $X=\{x_1,...,x_n\}$ of $\mathbb{R}^n$ there is a rotation that maps each $x_i$ into $V_{\tau(i)}$ where $\tau$ is a permutation.
If that is possible then what is the smallest $m$ so that the answer is affirmative. Does anybody know results of similar kind ? Thanks.
No. Look at $m = 1$ and $n = 2$. You're asking "is there a pair of lines through the origin in the plane with the property that for any independent 2-element set, some rotation takes the first vector to one line and the second vector to the other."
If there were, there'd be some angle between the two lines, say $\alpha$. Take any two vectors where the angle between them is neither $\alpha$ nor $\pi - \alpha$, and you cannot rotate these to like in the two specified lines, because rotation preserves angles.
My guess is that it's not true in any dimensions except $n = 0, 1$. Think about $m = 2$ and $n = 3$. The dihedral angles formed by the planes spanned by 2-element subsets of $x_1, x_2, x_3$ are invariant under rotation; if they don't match the dihedral angles (or their supplements) for the $V_i$, then you cannot find a rotation. I think this argument generalizes to all cases where $m = n - 1$, and it sure looks as if something like Plucker coordinates would suffice to handle cases of other codimensions.