Algebraic properties of R^n (or C^n)

Definition: The algebraic properties of (\mathbb{R}^n) (or (\mathbb{C}^n)) are the rules governing vector addition and scalar multiplication that make it a vector space.

For all (\mathbf{u}, \mathbf{v}, \mathbf{w} \in \mathbb{F}^n) and scalars (a, b \in \mathbb{F}):

1. Closure under addition: (\mathbf{u} + \mathbf{v} \in \mathbb{F}^n)
2. Commutativity: (\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u})
3. Associativity of addition: ((\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w}))
4. Additive identity: There exists (\mathbf{0} = (0,\ldots,0)) such that (\mathbf{u} + \mathbf{0} = \mathbf{u})
5. Additive inverse: For each (\mathbf{u}), there exists (-\mathbf{u}) such that (\mathbf{u} + (-\mathbf{u}) = \mathbf{0})
6. Closure under scalar multiplication: (a\mathbf{u} \in \mathbb{F}^n)
7. Distributivity (scalar over vector sum): (a(\mathbf{u} + \mathbf{v}) = a\mathbf{u} + a\mathbf{v})
8. Distributivity (scalar sum over vector): ((a + b)\mathbf{u} = a\mathbf{u} + b\mathbf{u})
9. Associativity of scalar multiplication: (a(b\mathbf{u}) = (ab)\mathbf{u})
10. Scalar identity: (1 \cdot \mathbf{u} = \mathbf{u})

These ten properties are the vector space axioms; verifying them for (\mathbb{F}^n) is straightforward from the component-wise definitions.