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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.

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  • $\begingroup$ 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 $V_i\supseteq V_k$. $\endgroup$ – user228113 Mar 1 '17 at 9:25
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    $\begingroup$ @G.Sassatelli Over an infinite field. $\endgroup$ – egreg Mar 1 '17 at 9:30
  • $\begingroup$ @egreg Whoops. True. $\endgroup$ – user228113 Mar 1 '17 at 9:39
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Hint: $(0,1,1)$ belongs to the first set, $(1,0,0)$ belongs to the second set; what about the sum?

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  • $\begingroup$ 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 $\endgroup$ – cnmesr Mar 1 '17 at 9:27
  • $\begingroup$ @cnmesr Does the sum belong to the union of the two sets? $\endgroup$ – egreg Mar 1 '17 at 9:29
  • $\begingroup$ 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. $\endgroup$ – cnmesr Mar 1 '17 at 9:33
  • $\begingroup$ @cnmesr Exactly; therefore… $\endgroup$ – egreg Mar 1 '17 at 10:31
  • $\begingroup$ Therefor the statement is false..? But why did you ask for the sum? $\endgroup$ – cnmesr Mar 1 '17 at 10:37

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