# The set $\left\{x \in \mathbb{R}^{3}|x_{1}=0\right\}\cup\left\{x \in\mathbb{R}^{3}| x_{2}=0\right\}$ is linear subspace of $\mathbb{R}^{3}$

Another task from a test-exam:

True or false? The set $\left\{x \in \mathbb{R}^{3} | x_{1}=0\right\} \cup \left\{x \in \mathbb{R}^{3}| x_{2}=0\right\}$ is a linear subspace of $\mathbb{R}^{3}$

I would say this statement is false because we have $x \in \mathbb{R}^{3}$ but only one element in both sets, I think it had to be a triple, something like $(0,1,2)$ to be true.

But as you can see I'm not sure at all and I would be very happy for some explanatory answers :)

Because we didn't get the solutions sadly.

• In general, the union of finitely many vector subspaces $V_1,\cdots, V_n$ is a vector subspace if and only if there is some $i$ such that for all $k$ it holds $V_i\supseteq V_k$. – user228113 Mar 1 '17 at 9:25
• @G.Sassatelli Over an infinite field. – egreg Mar 1 '17 at 9:30
• @egreg Whoops. True. – user228113 Mar 1 '17 at 9:39

## 1 Answer

Hint: $(0,1,1)$ belongs to the first set, $(1,0,0)$ belongs to the second set; what about the sum?

• I don't understand why you asking me this :s The sum should be $(1,1,1)$, right? And this would be in $\mathbb{R}^{3}$. But how can you say that $(0,1,1)$ belongs to first set, $(1,0,0)$ belongs to second? Because task says $x_{1}=0$ and other set $x_{2}=0$ Or your example doesn't belong to my task, it's another one explaining this one? I'm confused :P – cnmesr Mar 1 '17 at 9:27
• @cnmesr Does the sum belong to the union of the two sets? – egreg Mar 1 '17 at 9:29
• Hmm I think the union of the two sets would be $\left\{(0,1,1),(1,0,0)\right\}$ and the sum would be $(1,1,1)$ and this is not in the union. – cnmesr Mar 1 '17 at 9:33
• @cnmesr Exactly; therefore… – egreg Mar 1 '17 at 10:31
• Therefor the statement is false..? But why did you ask for the sum? – cnmesr Mar 1 '17 at 10:37