Let $\mathbb{h} \in \mathbb{Q}_{>0}^n$ a vector of positive rational numbers. Can you, please, help me prove that the vector space \begin{equation} Q = \lbrace \mathbf{q} \in \mathbb{Q}^n \,|\, \forall i, j \in \lbrace 1, 2,\dots, n \rbrace : q_i\,h_i = q_j\,h_j\rbrace \end{equation} is at least one dimensional?
If $\mathbb{q} \in \mathbb{Q}^n$ satisfies the previous equations than $\alpha\mathbb{q} \in \mathbb{Q}^n$ also satisfies the equations. However, I have trouble proving that something other than $\mathbf{0}$ vector satisfies the solution. Is it enough to say that a vector $q_1 = 1$ satisfies the equations?
I think my intuition is correct, but I am really having trouble formalizing it.