1. The problem statement, all variables and given/known data(adsbygoogle = window.adsbygoogle || []).push({});

A set of objects is given, together with operations of addition and scalar multiplication. Determine which sets are vector spaces under the given operations. For those that are not vector spaces, list all axioms that fail to hold.

(x,y,z) + (x',y',z') = (x+x',y+y',z+z') and k(x,y,z) = (kx,y,z)

2. Relevant equations

1. If u and v are objects in V, then u + v is in V.

2. u + v = v + u

3. u + (v + w) = (u + v) + w

4. There is an object 0 in V, called a zero vector for V, such that 0 + u = u+ 0 = for all un in V

5. For each u in V, there is an object -u in V, called a negative of u, such that u + (-u) = (-u) + u = 0.

6. If k is any scalar and u is any object in V, then ku is in V.

7. k(u + v) =ku + kv

8. (k + m)u = ku + mu

9. k(mu) = (km)(u)

10. 1u=u

3. The attempt at a solution

I have no idea what they are asking for, the back of the books say it fails one axiom which is: (k+m)u = ku +mu Which is axiom 8 in my book.

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# How can I tell if this is a vector space?

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