Originally Posted by

**mathguy25** There is nothing wrong with proving the method of contradiction, this is essentially the method you use when showing uniqueness. You basically assume two objects possess a certain characteristic and then show that these two objects must in fact be the same. Same reasoning as assuming they are different and finding a contradiction.

However, in your proof (1) you assumed u and w to be the zero vectors and used that assumption to conclude they were the same zero vector, seems a little too easy and too good to be true.

(2) is a good proof except use v^(-1) instead of -v unless you can prove that v^(-1) = -v

Proof of the uniqueness of inverses:

Let v be a vector in vector space V. Let v' and v'' be the inverse of v. Then by inverse additive axiom, v + v' = 0 and v + v'' = 0. Then v + v' = v + v''. Then add either v' or v'' to both sides. Then v' + v + v' = v' + v + v''. Then (v' + v) + v' = (v' + v) + v''. Then 0 + v' = 0 + v''. Then v' = v''. Thus, inverses are unique.

Proof that v^(-1) = -v.

v + v^(-1) = 0

-v + v + v^(-1) = -v + 0

(-1 + 1)v + v^(-1) = -v

0v + v^(-1) = -v

0 + v^(-1) = -v

v^(-1) = -v