The Four Subspace Problem, miniaturised

For any real number $k$ other than 0 or 1, define the quadruple $Q_k=(y=0,x=0,y=x,y=kx)$ . If $A$ defines an isomorphism from $Q_k$ to $Q_{k'}$ , then $A$ must be a scalar multiple of $I_2$ since $L_1,L_2$ and $L_3$ are invariant under $A$ . Thus $L_4$ is invariant under $A$ as well. We can conclude that $Q_k$ is isomorphic to $Q_{k'}$ only if $k=k'$ , showing that there are $\boxed{\text{infinitely many}}$ isomorphism classes of quadruples (and in fact overcountably many).