1. The problem statement, all variables and given/known data Let v, w, ∈ V. Prove that if v + w = 0, then w = -v. V is a complex vector space. 2. Relevant equations Axioms of a vector space. 3. The attempt at a solution So, this solution was pretty easy to come up with. My question is, have I proven that w = -v or have I simply proven that -v is unique? Let's see: Proof by contradiction: Suppose w and w' are additive inverses of v. w = w + 0 (By zero addition, axiom) w = w + (v + w') (by giving assumption) w = (w + v) + w' (by axiom 2 of vector spaces) w = w' (by giving assumption, w is additive inverse of v). Here, we have shown that the additive inverse of v is unique. But does this mean that w = -v as the question seems to state? Thanks.