**Theorem:** Vector addition in \(\mathbb{F}^n\) is both commutative and associative.

## Commutativity

For all \(\mathbf{u}, \mathbf{v} \in \mathbb{F}^n\):

\[\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}\]

**Proof:** \((\mathbf{u} + \mathbf{v})_j = u_j + v_j = v_j + u_j = (\mathbf{v} + \mathbf{u})_j\), since scalar addition is commutative.

**Example:** \((1, 2) + (3, 4) = (4, 6) = (3, 4) + (1, 2)\)

## Associativity

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

\[(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})\]

**Proof:** \(((\mathbf{u} + \mathbf{v}) + \mathbf{w})_j = (u_j + v_j) + w_j = u_j + (v_j + w_j) = (\mathbf{u} + (\mathbf{v} + \mathbf{w}))_j\), since scalar addition is associative.

**Consequence:** The order of adding three or more vectors does not matter; parentheses can be omitted: \(\mathbf{u} + \mathbf{v} + \mathbf{w}\) is unambiguous.