Thread: Vector Subspaces Union problem

Hello,

I am stuck with this problem.

prove that the union of two subspaces of the vector space $\displaystyle V$ is a subspace of $\displaystyle V$ if and only if one of the subspaces is contained in the other.

Thanks in advance for your help.

Can you prove that if one subspace is contained in the other, the union is a subspace? That should be relatively straight-forward.

Yes, I think I get that part. Is it right like this?
If $\displaystyle X$ and $\displaystyle Y$ are subspaces of $\displaystyle V$ and$\displaystyle X \subset Y$.
Then $\displaystyle X \bigcup Y = Y$.
So$\displaystyle X \bigcup Y$ is a subspace of $\displaystyle V$

But how would you the other way. Assuming first that the union of two subspaces, say $\displaystyle X$ and $\displaystyle Y$, of the vector space $\displaystyle V$. Then how do you prove that one of them is contained in the other?

Suppose that $\displaystyle X\cup Y$ is a subspace and $\displaystyle a\in X\setminus Y~\&~b\in Y\setminus X$.
We know that $\displaystyle a+b\in X\cup Y$.
What is wrong with that?

If $\displaystyle a \in X \backslash Y$ and $\displaystyle b \in Y \backslash X$ , then$\displaystyle X \backslash Y \neq \emptyset $ and $\displaystyle Y \backslash X \neq \emptyset$ . That would be assuming that the subspaces are not contained.... but I am still lost on how to continue.

Is it true that $\displaystyle a+b\in X\text{ or }a+b\in Y?$
Is it true that $\displaystyle -a\in X?$
There is a contradiction there. What is it?