Vector Subspaces Union problem

**akolman** · Aug 4th 2010, 01:01 PM

Hello,

I am stuck with this problem.

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

Thanks in advance for your help.

**Ackbeet** · Aug 4th 2010, 01:07 PM

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

**akolman** · Aug 4th 2010, 01:16 PM

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

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

**Plato** · Aug 4th 2010, 01:23 PM

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

**akolman** · Aug 4th 2010, 01:39 PM

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

**Plato** · Aug 4th 2010, 01:59 PM

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

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