Intersection Of Two Vector Subspaces

Statement:-

The intersection of any two sub-spaces of a vector space is also a subspace of the same vector space.

Proof:-

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

Since                         0єW1 and 0єW2

⇒                                    0єW1∩W2

Hence                           W1∩W2≠ф

Let                     a,bєF & α,βє W1∩W2

αє W1∩W2⇒  αєW1 & αєW2  and

βє W1∩W2⇒  βєW1  & βєW2

Hence                W1 is a subspace.

a,bєF      and   α,βєW1

⇒                       aα+bβєW1

Also                   W2 is a subspace

a,bєF  & α,βєW2⇒aα+bβєW2

Hence               aα+bβєW1,       aα+bβєW2

Since                 a,bєF  and α,βє W1∩W2

⇒                      aα+bβєW1∩W2

Thus W1∩W2 is a vector subspace.