# 10 Axioms of vector spaces

if u and v are objects in V, then u+v is in V

1

u+v = v+u

2

u+(v+w) = (u+v)+w

3

There is an object 0 in V called a zero vector for V,

Such that 0+u = u+0 = u

Such that 0+u = u+0 = u

4

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

5

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

6

k(u+v) = ku + kv

7

(k+m)u = ku+mu

8

k(mu) = (km)(u)

9

1u = u

10