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.