Just wondering. Suppose we some plane, any plane like [tex]S = \{ (x_1, x_2, x_3) \in F^{3} \ : \ x_1 + 5x_2 + 3x_3 = 0 \} [/tex] where F is either [tex]\mathbb{R}[/tex] or [tex]\mathbb{C}[/tex] . We know that S is a vector space (passes the origin).(adsbygoogle = window.adsbygoogle || []).push({});

We know that [tex](0,0,0)[/tex] is the additive identity and it should be unique by virtue of the field we're working with. But say we had any arbitrary vector [tex](a,b,c)[/tex]. Wouldn't [tex](-a, -b, -c)[/tex] or [tex](0, 3, -5)[/tex] also be counted as an additive identity as well?

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# Vector space question

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