Definition:Zero Vector
From ProofWiki
Definition
Let $\left({R, +_R, \times_R}\right)$ be a ring.
Let $\left({G, +_G}\right)$ be an abelian group.
Let $\left({G, +_G, \circ}\right)_R$ be an $R$-module.
The identity of $\left({G, +_G}\right)$ is usually denoted $\mathbf 0$, or some variant of this, and called the zero vector.
Note that on occasion it is advantageous to denote the zero vector differently, for example by $e$, or $0_V$ or $0_G$, in order to highlight the fact that the zero vector is not the same object as the zero scalar.
Zero Vector in $\R^n$
If the vector space in question is $\left({\R^n, +, \times}\right)_\R$, then the zero vector is:
- $\mathbf 0_{n \times 1} := \begin{bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end{bmatrix}$
where $0 \in \R$ is the zero scalar.
Sources
- Iain T. Adamson: Introduction to Field Theory (1964)... (previous)... (next): $\S 1.4$
- Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 26$
- C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 7.32$
