Suppose $O \in \mathbb{R}^{n \times r}$ with $r < n$ is a random matrix whose distribution is uniform on the set of $r \times n$ matrices such that $O'O = I_r$.
Is is true that the columns of $O$, marginally, are uniformly distributed on the unit sphere? If so, how would one prove this? Clearly, this is not true of the joint distribution of the columns.
Any insight would be appreciated!