Proving Inclusion of Vector Subspaces in W: A Scientific Approach

[Oster]

If U and V are vector subspaces of W and if U union V is also a subspace of W, how would i show that either U or V is contained in the other?

 

[Simon_Tyler]

Do you have an intuition for how it works?
(If you don't, think of say a line and plane in R^3. When is their union also a vector space?)
Then try translating that image into a proof.

 

[Petr Mugver]

If U and V were not contained in each other, you could find a basis of their union of the form

$$u_1,\dots,u_a,v_1,\dots,v_b,w_1,\dots,w_c$$

where the u's belong to U, the v's belong to V and the w's belong to the intersection (if this is not a zero dimensional space). Consider for example the vector

$$u_1+v_1$$

This vector can't belong to the union of U and V, because the basis above is composed of linearly independent vectors. So the union of U and V is not a vector space, a contraddiction.

 

[Oster]

@ Simon Tyler. yeah I've thought about those examples, like when the line is on the plane or one of the subspaces is the zero vector. But giving examples doesn't prove it..
Thanks a lot Petr. More questions to come soon =D

 

[Simon_Tyler]

@ Oster: I was hoping that thinking about that example would lead you to a proof like Petr's. Basically, you're question sounds like a homework problem -- which isn't helped by your small number of posts to these forums. These forums are not a place for homework help. I apologize if I've got you wrong and it as a genuine question.