Independent binary vectors over $\mathbb{R}$ are also independent over GF($2$)

Question

I am trying to prove that if a set of $m$ vectors over $\mathbb{R}^n$ — whose entries lie in $\{0,1\}$ — are linearly independent, then if we interpret them to be vectors over GF($2$), they would still be linearly independent. I formulate this problem as follows:

Let $\boldsymbol{v_1}, \boldsymbol{v_2} , \dots , \boldsymbol{v_m} \in \{ 0,1\}^n \subseteq \mathbb{R}^n$ be linearly independent vectors. Define the following matrix: $\Psi \triangleq \begin{pmatrix} \boldsymbol{v_1} & \boldsymbol{v_2} & \dots \boldsymbol{v_m} \end{pmatrix}$.
Then, $\exists \; $ an $m \times m$ minor of $\Psi$ such that its $\det(\cdot) \mod 2 \neq 0$ .

I need help proving this statement or an equivalent one in order to reach my objective.

Thanks.

Answer 1

How about

$$\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix} ?$$