How can I prove that v and w are equal if v + x = w + x?

[brandy]

Homework Statement

prove 0u=0 where u is in the vector space.

Homework Equations

the 10 various axioms for addition and scalar multiplication.

The Attempt at a Solution

pretty much just

(u+-u)u=0
or
(1-1)u=0
1u+-1u=0

and then i get stuck. i can prove that -1u=-u but that involves 0u-0

 

[Bearded Man]

Couldn't you let u = (a,b), then
0u
0(a,b)
(0a,0b)
(0,0)
0

 

[alexfloo]

No, Bearded Man. You made an assumption about "what vectors are" that doesn't follow from the axioms. Some vector spaces are described by components, but not all, and it's certainly not necessarily two-dimensional.

Try this:
u=1u=(1+0)u.

Distribute, and see where it takes you.

 

[aznboy]

Here you go Brandy:

0 x u = 0 + 0 x u
= -u + u + 0 x u
= -u + (1 + 0) x u
= -u + 1 x u
= -u + u
= 0

And maybe you can help me with this one.
v + x = w + x
Prove v = w