# Vector Spaces: Verify whether a set is a vector space

lawlbus
I was given a set of problems and answers, but not the solutions. This is one of the questions I've been having trouble in getting the correct answer.

## Homework Statement

Determine whether the set of vectors (u, v) is a vector space, where u + v = 0.

## Homework Equations

The 10 axioms of vector space.

## The Attempt at a Solution

The correct answer to this problem is TRUE, but I must be doing something wrong because I'm getting FALSE. I'm not quite sure if what I'm doing is right:

i) Both u and v are in V, and u + v is in V. (I'm not positive on this... is 0 always in V?)

ii) u + v = v + u (True)

iii) (u + v) + w = u + (v + w) (Not sure how to use the vector w in this one.. but it should be true if u + v = 0 as defined, right?)

iv) 0 + u = u + 0 = v (true)

v) u + (-u) = (-u) + u = u + v = 0 (true)

vi) c*u exists in V (false...? VERY UNSURE on this one! I thought it was false because of this!)

vii) c(u + v) = cu + cv (true)

viii) (c + d)u = cu + du (true)

ix) c(du) = (cd)u (true)

x) 1 * u = u (true)

I think my trouble is I'm not sure of the context on what V consists of.