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Definition of 0 vector

Created over 8 years ago, updated 3 days ago

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.