Linear automata over rational numbers field

We consider a class of linear automata over the field of rational numbers. In this class we prove there are no finite $K$-full systems and no finite $\Sigma$-full systems with infinite additive of special form. We construct an infinite $K$-basis and an infinite $\Sigma$-basis, and also an infinite $\Sigma$-full sistem which contains no $\Sigma$-basis.