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.
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.
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?
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.