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.
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