Let
$\alpha_1^1x_1+\ldots+\alpha_n^1x_n=1$
$\ldots$
$\alpha_1^mx_1+\ldots+\alpha_n^mx_n=1$
are equations in $\mathbb{F}_2^n$.(It is not system of equations). I am trying to prove that if every solution of $\beta_1x_1+\ldots+\beta_nx_n=1$ is solution of one of the equations then $(\beta_1,\ldots,\beta_n)$ is linear combination of $(\alpha_1^1,\ldots,\alpha_n^1),\ldots, (\alpha_1^m,\ldots,\alpha_n^m)$.
I can prove the inverse.
If $(\beta_1,\ldots,\beta_n)$ is linear combination of $(\alpha_1^1,\ldots,\alpha_n^1),\ldots, (\alpha_1^m,\ldots,\alpha_n^m)$ then every solution of $\beta_1x_1+\ldots+\beta_nx_n=1$ is solution of $\alpha_1^ix_1+\ldots+\alpha_n^ix_n=1$ for some $i$.