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.Copy