Union Of Two Subspace

Statement:-

The union of two subspace of a vector space is a subspace iff one each contained.

Proof:-

Let V be a vector space over the field F.Let W₁ and W₂ be its subspaces such that either

W₁⊆W₂                    or          W₂⊆W₁

If                   W₁⊆W₂          then

W₁∪W₂=W₂

Since,                   W₂ is a subspace.

Hence                  W₁∪W₂ is also a subspace.

Again,                if  W₂⊆W₁

then                     W₁∪W₂=W₁

Since,            W₁ being a subspace.

Hence           W₁∪W₂ will also be asubspace.

Conversally:-

Let W₁ and W₂ be subspaces of V such that W₁∪W₂ be also a subspace.

Let if possible W₁⊈W₂      or  W₂⊈W₁

Since     if         W₁⊈W₂

⇨    there exists      αєW₁ and α∉W₂

But                             α∊W₁∪W₂

If                                 W₂⊈W₁

⇨   there exists       β∊W₂ and β∉W₁

But                                  β∊W₁∪W₂

Hence                 W₁∪W₂ is a subspace.

Since              α∊W₁∪W₂   ,     β∊W₁∪W₂

⇨                    α+β∊W₁     or    α+β∊W₂

Let                  α+β∊W₁

Hence            α+β∊W₁   and  α∊W₁

⇨                   (α+β)-α∊W₁

⇨                      β∊W₂

Which is contradiction that  β∉W₁

Let                    α+β∉W₂

Since                α+β∊W₂   and β∊W₂

⇨                     (α+β)-β∊W₂

⇨                                α∊W₂

Which is contradiction that  α∉W₂

Hence                W₁⊆W₂    or  W₂⊆W₁                  Proved