How can we prove that the additive inverse -u is unique in a vector space?
Here's a hint. Please let me know if it's enough. Suppose v and w are additive inverses of u. What is v + u + w?
Let u be an arbitrary vector from some vector space. By the axioms of a vector space an additive inverse of u must exist. Suppose v and w are both additive inverses of u. By definition this implies: u + v 0 and u + w 0 u + v u + w [equivalence] v w [add (any) inverse u to both sides] Since v and w may be any possible choice of available inverse u's, all inverse u's are identical. i.e. the additive inverse of u is unique.
Suppose that B is an additive inverse of A, and if we also assume that C is another additive inverse of A, we will show that B C so inverses are unique. Because B and C are additive inverses of A we have: A + B 0 A + C 0 Now: B B + 0 (by the definition of 0) B + (A + C) (by the above, because C is an additive inverse of A) (B + A) + C (by associativity) 0 + C (by the above, because B is an additive inverse of A) C (by the definition of 0) So we have: B C And so there is a unique additive inverse for A
