- #1

RJLiberator

Gold Member

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## Homework Statement

Let

**v**,

**w, ∈**V. Prove that if

**v**+

**w**=

**0**, then

**w**=

**-v**.

V is a complex vector space.

## Homework Equations

Axioms of a vector space.

## The Attempt at a Solution

So, this solution was pretty easy to come up with.

My question is, have I proven that

**w**=

**-v**or have I simply proven that

**-v**is unique? Let's see:

Proof by contradiction:

Suppose

**w**and

**w'**are additive inverses of

**v.**

w=

w

**w**+

**0**(By zero addition, axiom)

**w**=

**w**+ (

**v**+

**w'**) (by giving assumption)

**w**= (

**w**+

**v**) +

**w'**(by axiom 2 of vector spaces)

**w**=

**w'**(by giving assumption, w is additive inverse of v).

Here, we have shown that the additive inverse of v is unique. But does this mean that

**w = -v**as the question seems to state?

Thanks.