Let K {\displaystyle {}K} be a field, and let V {\displaystyle {}V} be a K {\displaystyle {}K} -vector space. Let U , W ⊆ V {\displaystyle {}U,W\subseteq V} be linear subspaces of V {\displaystyle {}V} . Prove that the union U ∪ W {\displaystyle {}U\cup W} is a linear subspace of V {\displaystyle {}V} if and only if U ⊆ W {\displaystyle {}U\subseteq W} or W ⊆ U {\displaystyle {}W\subseteq U} .
