Theorem

Every element in a vector space has a unique additive inverse.

Proof

Suppose V is a vector space. Let ${latex.inlinev \in V}. Suppose w and w’ are additive inverses of v. Then we get:

${latex.inlinew = w + 0 = w + (v + w') = (w + v) + w' = 0 + w' = w'}

This gives us that w = w’ as desired. Note that the first equality holds due to the additive identity. The second due to w’ being an additive inverse. The third due to associativity in the vector space. The fourth due to commutativity and w being an additive inverse of v.

Counterexamples

Note that V being a vector space is crucial for this proof. Obviously, if it was not, there is no guarantee that an additive inverse exists, much less that it is unique. Furthermore, we rely on associative and commutative properties of addition in a vector space.