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Question: $S := \{x = (x_1, x_2, x_3, x_4, x_5) | x_1^2 + x_2^2 = x_3^2 + x_4^2 + x_5^2\}$

Is subset $S$ a valid subspace of $\mathbb{R}^5$?

I am not sure how to prove or disprove it if it's closed under addition. Because from:
$$(x_1+y_1)^2 + (x_2+y_2)^2 = (x_3+y_3)^2 + (x_4+y_4)^2 + (x_5+y_5)^2$$

I get:

$$x_1y_1+x_2y_2 = x_3y_3 + x_4y_4 + x_5y_5$$ and I don't know how to continue.

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2 Answers 2

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It is not closed under addition and therefore not a subspace: $(1,0,1,0,0), (0,1,1,0,0)\in S$ but their sum is not in $S$.

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    $\begingroup$ great, thank you so much! $\endgroup$
    – TTomi
    Nov 19, 2019 at 21:10
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To prove it's not closed under addition, you need just one counterexample.

Here's one for you to check: $(1,1,0,1,1),(1,1,1,0,1)\in S$ but is their sum?

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    $\begingroup$ thank you, great answer! $\endgroup$
    – TTomi
    Nov 19, 2019 at 21:16

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