Lattice of vectorial space

Question

Let $V$ be a vectorial space and $L(V)$ a lattice of the subspaces of $V$. Show that $L(V)$ is distributive if and only if $dim(V)=1$.

I don't even know how to start this problem. Any hint would be good. Thanks in advance

Answer 1

Hint: suppose there are two independent vectors $e_1, e_2 \in V$ and let $V_1 = \operatorname{span} \{ e_1 \}, V_2 = \operatorname{span} \{ e_2 \}, V_3 = \operatorname{span} \{ e_1 + e_2 \}$. Check that

$$( V_1 \vee V_2 ) \wedge V_3 \neq (V_1 \wedge V_3) \vee (V_2 \wedge V_3).$$