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.