Definition: The zero vector (or (\mathbf{0}) vector) in (\mathbb{F}^n) is the vector whose every component is zero:
[\mathbf{0} = (0, 0, \ldots, 0) \in \mathbb{F}^n]
Properties:
Additive identity: For any (\mathbf{v} \in \mathbb{F}^n): (\mathbf{v} + \mathbf{0} = \mathbf{v})
Scalar multiplication by zero:(0 \cdot \mathbf{v} = \mathbf{0}) for any (\mathbf{v})
Multiplication by zero vector:(a \cdot \mathbf{0} = \mathbf{0}) for any scalar (a)
Self-inverse:(\mathbf{0} = -\mathbf{0})
Examples:
In (\mathbb{R}^2): (\mathbf{0} = (0, 0))
In (\mathbb{R}^3): (\mathbf{0} = (0, 0, 0))
In (\mathbb{C}^4): (\mathbf{0} = (0+0i, 0+0i, 0+0i, 0+0i))
Role: The zero vector is the additive identity element of the vector space (\mathbb{F}^n) — it is one of the ten algebraic properties that define a vector space.
**Definition:** The *zero vector* (or *\(\mathbf{0}\) vector*) in \(\mathbb{F}^n\) is the vector whose every component is zero:
\[\mathbf{0} = (0, 0, \ldots, 0) \in \mathbb{F}^n\]
**Properties:**
- **Additive identity:** For any \(\mathbf{v} \in \mathbb{F}^n\): \(\mathbf{v} + \mathbf{0} = \mathbf{v}\)
- **Scalar multiplication by zero:** \(0 \cdot \mathbf{v} = \mathbf{0}\) for any \(\mathbf{v}\)
- **Multiplication by zero vector:** \(a \cdot \mathbf{0} = \mathbf{0}\) for any scalar \(a\)
- **Self-inverse:** \(\mathbf{0} = -\mathbf{0}\)
**Examples:**
- In \(\mathbb{R}^2\): \(\mathbf{0} = (0, 0)\)
- In \(\mathbb{R}^3\): \(\mathbf{0} = (0, 0, 0)\)
- In \(\mathbb{C}^4\): \(\mathbf{0} = (0+0i, 0+0i, 0+0i, 0+0i)\)
**Role:** The zero vector is the additive identity element of the vector space \(\mathbb{F}^n\) — it is one of the ten algebraic properties that define a vector space.