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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.

5xum
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1 Answers1

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You are on the right track. Try to be slightly more strict in writing what you are doing as it will help you think and explain your thoughts.

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 $t_1+t_2$?

5xum
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