We can define a vector space more formally. Given three vectors, v, w, and x, and two scalars, s and t, we begin by defining two operations:
1. Addition, v + w, which produces a sum.
2. Scalar multiplication, sv, which produces a scalar multiple.
In the most general definition of a vector space, these operations need not conform to the standard definitions we have explored in real vector spaces, Rn. What we do require is that the following 10 axioms are satisfied:
1. If v and w exist in V, then v + w exists in V as well.
2. v + w = w + v.
3. u + (v + w) = (u + v) + w.
4. There is a zero vector in V, 0, such that 0 + v = v + 0 = v for all v.
5. For every v, there is a negative of v, –v, such that v + (–v) = (–v) + v = 0.
6. For any scalar, s, and any vector in V, v, sv is also in V.
7. s(v + w) = sv + sw.
8. (s + t)v = sv + tv.
9. s(tv) = (st)v.
10. 1v = v.