Definition of equality of vectors

Definition: Two vectors (\mathbf{u}, \mathbf{v} \in \mathbb{F}^n) are equal if and only if they have the same size and their corresponding components are equal:

[\mathbf{u} = \mathbf{v} \quad \Longleftrightarrow \quad u_j = v_j \text{ for all } j = 1, 2, \ldots, n]

Requirements:

1. Same size: Both vectors must belong to the same space (\mathbb{F}^n)
2. Component-wise equality: Every corresponding pair of components must match

Examples:

• ((3, -1, 0) = (3, -1, 0)) ✓ — all components match
• ((3, -1, 0) \neq (3, -1, 1)) ✗ — third component differs
• ((3, -1) \neq (3, -1, 0)) ✗ — different sizes

Key consequence: Equality is defined component-wise, which means two vectors that look different algebraically might still be equal if they simplify to the same components. For instance, ((1+2, 3-3) = (3, 0)).