Intersection Of A Vector Space

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.

∵                      0єW1 and 0єW2  ⇒ 0єW1∩W2

∴                        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

∴               aα+bβєW1 & aα+bβєW2

∵                 a,bєF  and α,βє W1∩W2 ⇒  aα+bβєW1∩W2

Thus W1∩W2 is a vector subspace.