Proving that the Union of Two Non-Intersecting Subspaces is Not a Subspace

[lom]

V is a vectoric space.

W_1,W_2\subseteq V\\
W_1\nsubseteq W_2\\
W_2\nsubseteq W_1\\
prove that W_1 \cup W_2 is not a vectoric subspace of V.
i don't ave the shread of idea on how to tackle it

i only know to prove that some stuff is subspace

but constant mutiplication

and by sum of two coppies
this question here differs alot

 

[Mark44]

V is a vectoric space.

W_1,W_2\subseteq V\\
W_1\nsubseteq W_2\\
W_2\nsubseteq W_1\\
prove that W_1 \cdown W_2 is not a vectoric subspace of V.



i don't ave the shread of idea on how to tackle it

i only know to prove that some stuff is subspace

but constant mutiplication

and by sum of two coppies



this question here differs alot

Differs a lot from what? Is there any other information given in your problem? For example, you have that
W_1\nsubseteq W_2
and
W_2\nsubseteq W_1

but are you given anything about
W_1 \bigcap W_2
?

BTW, the term is "vector space" not vectoric space. No such word as vectoric.

 

[lom]

it differs by its pure theoretic way

i am used to prove that f(x)=0 is a subspace
by the constant multiplication and sum of two copies law
i only know that
<br /> W_1,W_2\subseteq V\\<br />

i have written all the given stuff

 

[Tedjn]

Since you are only given conditions on two subspaces, the only thing you can do next is look at the individual elements. The two conditions W_1 \not\subseteq W_2 and W_2 \not\subseteq W_1 imply the existence of what elements in these sets?

 

[lom]

W1 and W2 are foreign to each other
there is no intersection between them

 

[Tedjn]

Are you sure that there is no intersection between them? Let's take a step back. What is the actual condition that W_1 \not\subseteq W_2?

 

[lom]

its not only
<br /> W_1 \not\subseteq W_2<br />
its both
<br /> W_1 \not\subseteq W_2<br />
and <br /> W_2 \not\subseteq W_1<br />as for what you say:
<br /> W_1 \not\subseteq W_2<br />
means that all the members of W1 are not a part W2 group

 

[Tedjn]

No, it does not mean that all the members of W₁ are not in W₂. It means that there exists a member of W₁ that is not in W₂. Does that make sense? If so, where can you go from there?

 

[Mark44]

W1 and W2 are foreign to each other
there is no intersection between them

You can't conclude that from these two statements:
W_1\nsubseteq W_2\\
W_2\nsubseteq W_1\\

It's very possible that W₁ contains some elements that are in W₂, but other elements that aren't in W₂. Same thing for the other statement. That's why I asked if you were given that these two sets are disjoint. You said you weren't given that information, and now here you're saying that they are.