# Proving that the union of two vector sub spaces of the same space are not a sub space. [duplicate]

I hope that I translated it correctly, correct me if I'm wrong :)

Let $V_1$ and $V_2$ be sub spaces of $L$.

Prove that $V_1\cup V_2$ is a vector sub space if and only if $V_1 \subseteq V_2$ or $V_2\subseteq V_1$

I was thinking of assuming it is correct and taking $2$ vectors from each subspace such that: $$t_1 \notin V_2 , t_2 \notin V_1 \\ t_1,t_2 \in V_1 \cup V_2$$ Then : $$t_1 + t_2 ∈ V1$$

This is where I am unsure how to explain my moves...

The teacher said it is incorrect, do you guys have any hints then, perhaps on how to refine a better mathematical 'language of reasoning' too? I am new at this whole math world :)

Many thanks.

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## marked as duplicate by Zev Chonoles, user127.0.0.1, Davide Giraudo, Daryl, FabianFeb 26 '14 at 9:38

What you want to prove is that if neither $V_1\subseteq V_2$ nor $V_2\subseteq V_1$ holds, then $V_1\cup V_2$ is not a vector space. For this, you have correctly chosen $t_1\in V_1\setminus V_2$ and $t_2\in V_2\setminus V_1$. Then you have chosen to calculate $t_1+t_2$.
The mistake you made here is writing $t_1+t_2\in V_1$, which is not something you have proven. Try to think about what you can prove about $t_1+t_2$. Hint: $t_1$ and $t_2$ are both elements of $V_1\cup V_2$. If $V_1\cup V_2$ is a vector space, what does that tell you about $t_1+t_2$?