Short Proof regarding elements of a vector space

[srfriggen]

Homework Statement

Let V be a vector space and v, w two elements of V. If v+w=O, show that w=-v

Homework Equations

The Attempt at a Solution

This is my attempt:

1. v+w=O
2. v+(-v)=O
3. v+w=v+(-v). Subtracting v from both sides yields:
4. w=-v



The solution manual has this as the correct answer:

-w=-w+O=-w+(v+w)=v+w-w=v



Is my attempt correct as well?

 

[HallsofIvy]

Yes, your proof is perfectly valid.

 

[gopher_p]

What properties of vector spaces are you permitted to use? I'm mostly concerned with going from step 3 to step 4. If you are permitted to add to both sides of an equality, presumably you could have done that from the start:

v+w=0 (Given)
-v+(v+w)=-v+0 (Left addition on both sides of an equality: this is the property I'm worried about you using)
(-v+v)+w=-v+0 (Associativity of +: the book solution uses it)
0+w=-v+0 (Definition of -v)
w=-v (Definition of 0)

Speaking of subtraction, you need to be careful with that as well. It's possible you haven't yet defined what v-w means. It's probably more appropriate to say "adding the additive inverse", like you wrote in step 2.

You need to be really careful when proving "obvious" statements that you only use the properties that are assumed or that you've already proved. It helps to identify the exact property that you're using every step of the way.

Note that you also use transitivity of = (in 1 & 2 to 3), but so does your book. The book also also uses commutativity of +.