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Vector sum/addition is commutative and associative

Created over 8 years ago, updated 3 days ago

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.