# How to end the proof

1. Oct 13, 2008

### soul

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: Oct 13, 2008
2. Oct 13, 2008

### Defennder

??? 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.

3. Oct 13, 2008

### soul

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.

4. Oct 13, 2008

### HallsofIvy

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

5. Oct 13, 2008

### d_leet

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

6. Oct 13, 2008

### JasonRox

The axioms say there must exist a zero vector. It does not say it is unique or must be unique. You prove that it is unique if there exists such a vector.

7. Oct 15, 2008

### Jang Jin Hong

It is impossible to reply. You do not said definition of the vector space, and do not said about preceding procedure.