Hello,
Is this right? I know this is trickier than it looks because you are only allowed to assume the axioms of vector spaces alone, nothing else.
It is right but, imo, ugly. I prefer the following (be sure you can write down the axiom on which each step relies):
$\displaystyle c\cdot 0=c(0+0)=c\cdot 0+c\cdot 0\Longrightarrow c\cdot 0+(-c\cdot 0)=c\cdot 0+c\cdot 0+(-c\cdot 0) \Longrightarrow 0 = c\cdot 0 +0=c\cdot 0$ QED.
Tonio
only one quibble, tonio....i believe you should have written -(c0) rather than (-c)0.
yes, these are in point of fact equal, but proving that uses the fact that 0v = 0, for any v, which is somewhat of a detour.
(it's just a misplaced parenthesis, i'm sure you meant -(c0)).