Suppose that $S$ and $S_0$ are two systems of linear equations over a field $k$. Show that $(S) = (S_0)$ if and only if $Sol(S; k) = Sol(S_0; k)$.
Note that $(S)=(S_0)$ if and only if $Sol(S; K) = Sol(S_0; K)$ for any $k$-algebra $K$. So the only if direction is easy. But I don't know how to solve the converse.