# How to end the proof

Hi eveyone,

I was trying to prove that for the vector spaces, there is a unique vector that satisfy "u + 0 = u" and I used contradiction technique. The last point that I reached is $$u + 0_1 = u + 0_2$$. However, I don't know whether I can say $$0_1 = 0_2$$ after this statement or there are some other operations that I must do (like this statement needs a proof as well?).

Thank you.

Last edited:

Defennder
Homework Helper
??? Why do you need to prove that? Isn't that an axiom that every vector space has to satisfy? Namely that every vector space has a unique zero vector? What axioms do you start off with?

You can "prove" this by noting that along with your last step, -u also exists in the same vector space.

Sorry, I guess I wrote my question wrong. I was trying to prove there is a unique identitiy element in summation and what I did is to select two different vectors and at the end of it, to show they are the same. I used the axiom in the question.

HallsofIvy
Homework Helper
Uniqueness of the identity is an axiom in groups but can be proved in a vector space.

Uniqueness of the identity is an axiom in groups but can be proved in a vector space.

Is this true? Almost every group theory book I have looked at proves the uniqueness of the identity as a theorem.

JasonRox
Homework Helper
Gold Member
Uniqueness of the identity is an axiom in groups but can be proved in a vector space.