Zermelo
Junior Member
- Joined
- Jan 7, 2021
- Messages
- 148
Hi, I have a little problem with one of the problems from my Linear Algebra 1 class.
The exercise goes like this:
Let U and W be vector subspaces of the vector space V.
Prove:
If U [MATH] \bigcup [/MATH] W is a vector subspace of V [MATH]\Longrightarrow[/MATH] U [MATH]\subseteq[/MATH] W or W [MATH]\subseteq[/MATH] U.
The proof goes like this:
Let's suppose the opposite, that U isn't a subset of W and W isn't a subset of U. That means that there exist elements u [MATH]\in[/MATH] U\W and w [MATH]\in[/MATH] W\U.
u + w [MATH]\in[/MATH] U [MATH] \bigcup [/MATH] W [MATH]\overset{don't get this} \Longrightarrow[/MATH] w [MATH]\in[/MATH] U or u [MATH]\in[/MATH] W, which is a contradiction ete etc, qed.
My question is, is it possible to have an element u from U\W and w from W\U, such that u + w is in U? I can't prove that this isn't possible, and my professor didn't prove it either, this seems trivial but I can't find a formal proof that this isn't possible.
It is clear that for all u and w, if u and w [MATH]\in[/MATH] U [MATH]\Longrightarrow[/MATH] u + w [MATH]\in[/MATH] U, because U is a vector space and that means (U, +) is an Abel group. But why does the opposite work? Here in the proof, we have u+w [MATH]\in[/MATH] U [MATH]\Longrightarrow[/MATH] u [MATH]\in[/MATH] U.
I am looking for a formal proof that it isn't possible for the sum of two elements from different vector spaces to belong to one of those vector spaces, or a formal proof of the step that I don't understand above. I hope you can understand what I mean, English isn't my first language Thanks in advance
The exercise goes like this:
Let U and W be vector subspaces of the vector space V.
Prove:
If U [MATH] \bigcup [/MATH] W is a vector subspace of V [MATH]\Longrightarrow[/MATH] U [MATH]\subseteq[/MATH] W or W [MATH]\subseteq[/MATH] U.
The proof goes like this:
Let's suppose the opposite, that U isn't a subset of W and W isn't a subset of U. That means that there exist elements u [MATH]\in[/MATH] U\W and w [MATH]\in[/MATH] W\U.
u + w [MATH]\in[/MATH] U [MATH] \bigcup [/MATH] W [MATH]\overset{don't get this} \Longrightarrow[/MATH] w [MATH]\in[/MATH] U or u [MATH]\in[/MATH] W, which is a contradiction ete etc, qed.
My question is, is it possible to have an element u from U\W and w from W\U, such that u + w is in U? I can't prove that this isn't possible, and my professor didn't prove it either, this seems trivial but I can't find a formal proof that this isn't possible.
It is clear that for all u and w, if u and w [MATH]\in[/MATH] U [MATH]\Longrightarrow[/MATH] u + w [MATH]\in[/MATH] U, because U is a vector space and that means (U, +) is an Abel group. But why does the opposite work? Here in the proof, we have u+w [MATH]\in[/MATH] U [MATH]\Longrightarrow[/MATH] u [MATH]\in[/MATH] U.
I am looking for a formal proof that it isn't possible for the sum of two elements from different vector spaces to belong to one of those vector spaces, or a formal proof of the step that I don't understand above. I hope you can understand what I mean, English isn't my first language Thanks in advance