# Thread: A U B is not a subspace

1. ## A U B is not a subspace

Hello everyone!

I'm after this linear algebra question:
$A \text{ and }B \text{are subspaces of a vector space }V. \text{ show that }A\cup B \text{ are a subspace of }V \Leftrightarrow A\subseteq B \text{ or } B \subseteq A.$

Proving that $0\in V$ is easy, howeve I'm not sure how to write down the proof for the other 2 axioms on paper though my professor cleared the entire matter up.

Thanks!

2. Originally Posted by rebghb
Hello everyone!

I'm after this linear algebra question:
$A \text{ and }B \text{are subspaces of a vector space }V. \text{ show that }A\cup B \text{ are a subspace of }V \Leftrightarrow A\subseteq B \text{ or } B \subseteq A.$

Proving that $0\in V$ is easy, howeve I'm not sure how to write down the proof for the other 2 axioms on paper though my professor cleared the entire matter up.

Thanks!

Take $a\in A\,,\,\,b\in B$ , now: where is $a+b$ ?

Tonio

3. a + b is neither in A nor in B, I know, it's in A U B, but how to prove that!

4. Originally Posted by rebghb
a + b is neither in A nor in B, I know, it's in A U B, but how to prove that!
That makes no sense. If a+ b is in $A\cup B$ it must be in either A or B (or both). That's how $A\cup B$ is defined!

I think that you mean "if $A\cup B$ is a subspace, then, for all a in A, b in B, a+ b must be in $A\cup B$ and so must be in either A or B."