# Proving v=w using Vector Space Axioms

• aznboy
In summary, the problem involves proving that v and w are equal using the axioms of an arbitrary vector space, where v, w, and x are all part of V. The key to solving the problem is to use the fact that x has an additive inverse, and to incorporate the 0 vector into the equation.
aznboy

## Homework Statement

V is an arbitrary vector space and v,w,x are part of V such that v + x = w = x

Use vector space axioms to prove v = w

I've looked at the axioms for an hour and can not get any lead to start this question.

You mean $v + x = w + x$ right?

Well, $x \in V$, right? Doesn't it have an additive inverse?

yeah it satifies the 10 vector axioms, but i can't seem to muck around with the axioms to solve the problem.

Yeah sorry i meant v + x = w + x

I know it has something to do with the 0 vector. ie 0 = +x -x

When i incorporate that into the eqn i get v +x -x +x = w +x -x +x ... which I am not sure if it helps

Last edited:

## 1. What are the vector space axioms?

The vector space axioms are a set of properties that define the rules for vector operations and their properties. These axioms include properties such as closure, commutativity, associativity, distributivity, and the existence of an identity element.

## 2. How do these axioms relate to proving v=w?

The vector space axioms provide a framework for proving that two vectors, v and w, are equal. By using these axioms, we can show that all the properties of v are also true for w, and vice versa, which ultimately proves that v and w are equal.

## 3. Can v=w be proven using other methods?

Yes, v=w can also be proven using other methods such as algebra or geometry. However, the use of vector space axioms is a more general and rigorous approach to proving vector equality.

## 4. What are some examples of vector space axioms being used to prove v=w?

One example is using the distributivity axiom to show that scalar multiplication of a vector is associative, and then using this to prove that v=0 if and only if w=0. Another example is using the closure axiom to show that if v=w, then v+w=w+v.

## 5. Are there any limitations to using vector space axioms to prove v=w?

While the vector space axioms provide a powerful tool for proving vector equality, they do have limitations. Some vector spaces may not follow all of the axioms, making it impossible to use this method. Additionally, the proof may become complex for larger vector spaces.

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