Created almost 9 years ago, updated 5 months ago
ID 23787
A− A+
Theorem: Vector addition in ( \mathbb{F}^n) is both commutative and associative.
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))
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.
**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.Copy