Answer to Question #224410 in Linear Algebra for Unknown346307

Question

Prove that V₁$\cap$ V₂ is a subspace of W if and only if V₁$\subseteq$ V₂ or V₂⊆V₁ in which V₁ and V₂ are subspaces of vector space W.

Answer 1

Let's prove that V1$\bigcap$ V2 is subspace in W in all cases without any additional conditions.

Let x,y$\in$ V1$\bigcap$ V2 (any two elements in intersection) and $\alpha,\beta\in$R (any two real coefficients).We must prove that a linear combination x*$\alpha$ +y*$\beta$$\in$ V1$\bigcap$ V2 too. By definition of intersection we have x,y$\in$ V1 and x,y$\in$ V2, besides, because V1,V2 are subspaces in W we have x*$\alpha$ +y*$\beta$$\in$ V1 and x*$\alpha$ +y*$\beta$$\in$ V2 and using again definition of intersection we have in result x*$\alpha$ +y*$\beta$$\in$ V1$\bigcap$ V2 and this means that the proof is finished.