Let $M$ be an $n \times m$ matrix over $\mathbb{F}_2$ with no repeated columns, and suppose that $m \leq 2^{n-1}$: *i.e.*, it is possible to have a matrix with fewer rows that still has $m$ unique columns.

Is it always possible to find such a smaller matrix by taking a linear combination of the rows of $M$?

For example, $M=\pmatrix{1 & 0 & 0 \\ 0 & 1 &0 \\ 1 & 0 & 1}$ has distinct columns, but we can find a smaller matrix with distinct columns by taking $$\pmatrix{1 & 1 & 0 \\ 0 & 0 &1 }\pmatrix{1 & 0 & 0 \\ 0 & 1 &0 \\ 1 & 0 & 1} = \pmatrix{1 & 1 & 0 \\ 1 & 0 &1 }.$$