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.
**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.