Vector Spaces

Question 1

Definition of Vector Spaces

Answer 1

A vector space is an nonempty set V of objects, called vectors, on which are defined two operations, called addition, and multiplication by scalars (real numbers), subject to the ten axioms (or rules) listed below. The axioms must hold for all vectors u, v, and w in V and for all scalars c and d.

1. The sum of u and v, denoted by u+v, is in V.
2. u + v = v + u
3. (u+v) + w = u + (v + w)
4. There is a zero vector 0 in V such that u + 0 = u
5. For each u in V, there is a vector -u in V such that u + (-u) =0
6. The scalar multiple of u by c, denoted by cu is in V.