Let $v_1, . . . , v_m$ be vectors with $n$ $(0,1)$-entries. Prove that if these vectors are linearly independent over $\mathbb{F}_p$ for some prime number $p$ then they are linearly independent over $\mathbb{Q}$.
My attempt:
If we have a $n$ by $m$ matrix $A$ whose column vectors are $v_1, . . . , v_m$, and they are linearly independent over the field $\mathbb{F_p}$ where p is a prime number. That implies $det(A)$ mod $p \neq 0$.
Then in the field $\mathbb{Q}$, there is also $det(A)$ mod $p \neq 0$, so the column vectors are linearly independent over the field $\mathbb{Q}$.
I am not sure it is ok.
Best Answer
Suppose that the vectors are $\Bbb Q$-linearly dependent. Clearing denominators, obtain a non-trivial $\Bbb Z$-linear combination of these vectors that equals zero. Now consider the residues modulo $p$.