# Vector Subspaces Union problem

• Aug 4th 2010, 01:01 PM
akolman
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.

• Aug 4th 2010, 01:07 PM
Ackbeet
Can you prove that if one subspace is contained in the other, the union is a subspace? That should be relatively straight-forward.
• Aug 4th 2010, 01:16 PM
akolman
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?
• Aug 4th 2010, 01:23 PM
Plato
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?
• Aug 4th 2010, 01:39 PM
akolman
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.
• Aug 4th 2010, 01:59 PM
Plato
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?