# Vector Subspaces Union problem

• Aug 4th 2010, 02:01 PM
akolman
Vector Subspaces Union problem
Hello,

I am stuck with this problem.

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

• Aug 4th 2010, 02:07 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, 02:16 PM
akolman
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?
• Aug 4th 2010, 02:23 PM
Plato
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?
• Aug 4th 2010, 02:39 PM
akolman
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.
• Aug 4th 2010, 02:59 PM
Plato
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?