Suppose we have a orthonormal matrix $ V \in \mathcal{R}^{n \times n}$ (i.e. $V V^T = I$). The columns of the $V$ matrix are othogonal unit vectors.
Build the matrix $V_k$ composed of the first $k < m$ columns of $V$.
Then take a vector $x \in \lbrace 0, 1\rbrace^n$ and multiply: $y = xV_k V_k^T$
Is it possible that for some $i \in \lbrace 1, 2, \dots, n \rbrace$ we have: $y_i > 1$ ?