Definition of vector sum/addition

Definition: The sum (or addition) of two vectors (\mathbf{u}, \mathbf{v} \in \mathbb{F}^n) is the vector obtained by adding corresponding components:

[\mathbf{u} + \mathbf{v} = (u_1 + v_1,; u_2 + v_2,; \ldots,; u_n + v_n)]

Component-wise formula: ((\mathbf{u} + \mathbf{v})_j = u_j + v_j) for each (j = 1, 2, \ldots, n).

Requirements: Both vectors must have the same size (belong to the same (\mathbb{F}^n)).

Examples:

• In (\mathbb{R}^3): ((1, 2, 3) + (4, -1, 0) = (5, 1, 3))
• In (\mathbb{C}^2): ((1+i, 2-i) + (3, i) = (4+i, 2))

Geometric interpretation ((\mathbb{R}^2)): Vector addition corresponds to the parallelogram rule — place the tail of (\mathbf{v}) at the head of (\mathbf{u}); the sum is the diagonal from the origin.

Properties: Vector addition is closed in (\mathbb{F}^n), commutative, associative, has an identity element (\mathbf{0}), and every vector has an additive inverse.