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.

 

[radou]

I'm not sure what's your axiom vi) supposed to mean?