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:
Same size: Both vectors must belong to the same space (\mathbb{F}^n)
Component-wise equality: Every corresponding pair of components must match
Examples:
((3, -1, 0) = (3, -1, 0)) ✓ — all components match
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)).
**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, \((1+2, 3-3) = (3, 0)\).