Vector Space

A vector space is a set V of elements called vectors, having operations of addition and scalar multiplication defined on it that satisfy the following conditions. (u, v, and w are arbitrary elements of V, and c and d are scalars.)

Closure Axioms

1. The sum u + v exists and is an element of V. (V is closed under addition.)

2. cu is an element of V. (V is closed under scalar multiplication.)

Addition Axioms

3. u + v = v + u (commutative property)

4. u + (v + w) = (u + v) + w (associative property)

5. There exists an element of V, called the zero vector, denoted 0, such that u + 0 = u.

6. For every element u of V there exists an element called the negative of u, denoted −u, such that u + (−u) = 0.

Scalar Multiplication Axioms

7. c(u + v) = cu + cv

8. (c + d)u = cu + du

9. c(du) = (cd)u

10.1u = u