mastboyx

So heres a tough one, dunno if u guys could help me:

If U and W are linear subspaces of a linear space V. Demonstrate that if every matrix that belongs to V belongs to U or W, then U = V or W = V...

__________________
Ignorance Is Bliss

KnifeMissile

This may sound weird but your question doesn't seem so difficult, nor do I understand it. Let me rephrase your question in a manner that makes sense to me and tell me if it's the same question. It sounds like you've got some terminology mixed up...

Let U and W be subspaces of a vector space V. Prove that if every vector in V belongs to U or W, then U = V or W = V.

Is this right?
It would also help to know what it is you can use. Do you know about vector bases, spanning, and dimensions?

mastboyx

yeah i guess thats exactly right, and yeah u are supposed to solve it with the whole spanning concept and dimensions and stuff, the thing is I speak spanish and i had a bit of trouble translating these terms, but yeah what u said sounds pretty much what my question is about, , any idea how to solve it?

__________________
Ignorance Is Bliss

KnifeMissile

Okay, cool. Your english is very good for a non-native speaker.

If you've been reading this forum for a while, you might have learned that I don't generally just prove other people's theorems for them, especially if it's an assignment! I honestly want you to learn this stuff on your own, right? On the other hand, if you're simply stuck, staying stuck won't teach you anything so here's a compromise. I'll prove to you a really useful lemma that you can use to prove your theorem, okay? Here it is...


lemma: If U and W are vector spaces and the union of U and W is also a vector space, then one of them must be a subspace of the other.

proof: Suppose that neither U nor W are subspaces of each other. Then, by definition, there must exist vectors u from U and w from W such that w is not a member of U and u is not a member of W.

Because the union of U and W is a vector space, the sum u + w must also be in this union. By definition, this means that u + w is either in U or in W. However, if it's in U then u + w - u must also be in U, but u + w - u = w, which is not supposed to be in U! This is a contradiction, so u + w must be in W.

Folowing the same logic the other way, you can deduce that u + w is not in W either, so the whole situation is a contradiction. If no such u and w can exist, then one vector space must necessarily be a subset, and therefore a subspace, of the other.
QED.


I hope this helps! If it doesn't, come back and I'll show you how this lemma applies to your problem...

mastboyx

oh, no thanks it does help, and yeah i was a bit stuck with the whole thing, i had no idea how to approach the problem, and its not an assignment, im taking this class but the thing is its a masters degree class, and im still at bachelor classes, so im a bit behind on some subjects, thats why i was a bit confused with this thing he "explained" it to us but he said there was another way and that we should look it up, but not as an assignment but as self study.. so thanks so much for the help

__________________
Ignorance Is Bliss

KnifeMissile

Hey, I'm glad that helped!

It's interesting that you were so incredulous that anyone here could help you. There are some very good mathematicians on this board...

mastboyx

yeah i can see good know people to learn from thanks!!

__________________
Ignorance Is Bliss

mastboyx

*good to know*

__________________
Ignorance Is Bliss