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.
∴ 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.
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