Let W {\displaystyle {}W} be an n {\displaystyle {}n} -dimensional K {\displaystyle {}K} -vector space ( K {\displaystyle {}K} a field), and let U , V ⊆ W {\displaystyle {}U,V\subseteq W} be linear subspaces of dimension dim K ( U ) = r {\displaystyle {}\dim _{K}{\left(U\right)}=r} and dim K ( V ) = s {\displaystyle {}\dim _{K}{\left(V\right)}=s} . Suppose that r + s > n {\displaystyle {}r+s>n} holds. Show that U ∩ V ≠ 0 {\displaystyle {}U\cap V\neq 0} .
