A community for students.
Here's the question you clicked on:
 0 viewing
gorica
 3 years ago
Let W, S, T be subspaces of finitedimensional vector space such that
S∩T=S∩W
S+T=S+W, and W is subset of T.
Prove that W=T.
gorica
 3 years ago
Let W, S, T be subspaces of finitedimensional vector space such that S∩T=S∩W S+T=S+W, and W is subset of T. Prove that W=T.

This Question is Closed

abb0t
 3 years ago
Best ResponseYou've already chosen the best response.0Definition of Composition?

malevolence19
 3 years ago
Best ResponseYou've already chosen the best response.0I'm not sure of a formal proof but if you think about it the only way that S+T=S+W is if they are the same. If they were not the same that would mean there is an element of T not in W or an element of W not in T. If not you'd have something like {1,2}+{3,4,5}={1,2}+{3,4,6} implies {1,2,3,4,5}={1,2,3,4,6} which is not true obviously. It is still true that their intersections can be equal. {1,2} intersect {3,4,5} is the same as {1,2} intersect {3,4,6} (in this case the empty set). I think that the fact of their intersections with S are equal is superfluous information but I didn't care for proofs too much.

gorica
 3 years ago
Best ResponseYou've already chosen the best response.0actually, I have to prove that T is subset of W, using what is given, and that will imply that T=W since I already have given that W is subset of T.

AddemF
 3 years ago
Best ResponseYou've already chosen the best response.1Consider any arbitrary element of T, call it t. We want to show that t is in W. Suppose that t is not in W and let's try to get a contradiction. If t is not in W then it is still in S+T which is the same as S+W and so there exist some s in S and w in W such that t = s+w. But since we know W is a subset of T, then this w is in T. For that reason, let's rename it t1, to show that it is actually in T. That means t = s+t1. And since the intersection of S and T is a subspace, it is closed under addition. Therefore s+t1 is an element of the intersection of S and T, and since this is the same thing as t, then t is in the intersection of S and T. But the intersection of S and T is the same as the intersection of S and W. This means t is in W. This contradicts our initial assumption. Therefore t is in W. By definition of set inclusion, T is a subset of W.
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.