Definition:Zero Vector

Definition

Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {G, +_G}$ be an abelian group.

Let $\struct {G, +_G, \circ}_R$ be an $R$-module.

The identity of $\struct {G, +_G}$ 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 $\R^n$

Let $\struct {\R^n, +, \times}_\R$ be a real vector space.

The zero vector in $\struct {\R^n, +, \times}_\R$ is:

$\mathbf 0_{n \times 1} := \begin {bmatrix} 0 \\ 0 \\ \vdots \\ 0 \end {bmatrix}$

where $0 \in \R$.

Also known as

The zero vector is also sometimes known as the null vector.

Sources

• 1964: Iain T. Adamson: Introduction to Field Theory ... (previous) ... (next): $\S 1.4$
• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): $\S 26$
• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 22$: Fundamental Definitions: $2.$
• 1969: C.R.J. Clapham: Introduction to Abstract Algebra ... (previous) ... (next): Chapter $7$: Vector Spaces: $\S 32$. Definition of a Vector Space
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: zero vector (null vector): 2.