I'm working on the following problem:
Let $W$ be a subspace of the vector space $\mathbb{K}^n$, where $\mathbb{K}$ is a field of characteristic $0$. Let $L$ denote the set of supports of all vectors in $W$, ordered by reverse inclusion. Show that $L$ is a geometric lattice.
**I will mention that for any $v=(v_1, \ldots , v_n)$, $\,$ $\text{supp}(v)=\left\{i: v_i\neq 0 \right\}$.
First I need to show that $L$ is semimodular, so I will show that if $\text{supp}(v)$ and $\text{supp}(u)$ both cover $\,$ $\text{supp}(v) \wedge \text{supp}(u)$ $\,$ then $\,$ $\text{supp}(v) \vee \text{supp}(u)$ $\,$ cover both $\,$ $\text{supp}(v)$ $\,$ and $\,$ $\text{supp}(u)$.
Here is my issue: I want to safely say that since $\text{supp}(v) \subseteq [n]$ that $\,$ $\text{supp}(v) \vee \text{supp}(u)= \text{supp}(v) \cup \text{supp}(u)$ $\,$ and $\,$ $\text{supp}(v) \wedge \text{supp}(u)= \text{supp}(v) \cap \text{supp}(u)$. However, how do I know that $L$ is closed under unions and intersections to begin with? Perhaps I'm wrong and the join and meet of any two elements in $L$ is some other set. Help me, please! Thank you!