Zero vector

In mathematics, the zero vector is a special vector of a vector space , namely the uniquely determined neutral element with regard to vector addition . Examples of zero vectors are the number zero , the zero matrix and the zero function . In a scalar product space, the zero vector is orthogonal to all vectors of the space. In a normalized space it is the only vector with norm zero. Each sub-vector space of a vector space contains at least the zero vector, the smallest sub-vector space being the zero vector space . The zero vector is used to define some of the key terms in linear algebra such as linear independence , base, and kernel . It plays an important role in the solution structure of linear equations .

definition

The zero vector of a vector space is the uniquely determined vector for which ${\displaystyle V}$ ${\displaystyle 0_{V}\in V}$

${\displaystyle v+0_{V}=0_{V}+v=v}$

holds for all vectors . It is therefore the neutral element with regard to vector addition . ${\displaystyle v\in V}$

notation

The zero vector is usually denoted by means of the number zero through , or simply simply . The zero vector, however, is generally different from the zero element of the scalar field of the vector space, which is also represented by. If there is a risk of confusion, the zero vector is labeled with and the scalar zero with . Occasionally the zero vector is also notated by , or as a small o . ${\displaystyle {\vec {0}}}$ ${\displaystyle \mathbf {0} }$ ${\displaystyle 0}$ ${\displaystyle K}$ ${\displaystyle 0}$ ${\displaystyle 0_{V}}$ ${\displaystyle 0_{K}}$ ${\displaystyle {\vec {o}}}$ ${\displaystyle \mathbf {o} }$ ${\displaystyle {\mathfrak {o}}}$

As the only vector of the Euclidean plane , the zero vector cannot be graphically represented by an arrow , since neither direction nor length can be assigned to it.

Examples

• In the vector space of real numbers , the zero vector is the number and thus equal to the zero of the scalar field. ${\displaystyle \mathbb {R} }$ ${\displaystyle 0}$

• In the vector space of complex numbers , the zero vector is the number and thus also corresponds to the scalar zero. ${\displaystyle \mathbb {C} }$ ${\displaystyle 0+0i}$

• In the coordinate space , the zero vector is the n -tuple consisting of the zero elements of the body . ${\displaystyle K^{n}}$ ${\displaystyle (0_{K},\ldots ,0_{K})}$ ${\displaystyle K}$

• In matrix space , the zero vector is the zero matrix , the elements of which are all equal . ${\displaystyle K^{m\times n}}$ ${\displaystyle 0_{K}}$

• In the sequence space the zero vector is the sequence and not to be confused with the concept of the zero sequence . ${\displaystyle {\mathbb {K} }^{\mathbb {N} }}$ ${\displaystyle (0_{\mathbb {K} },0_{\mathbb {K} },\ldots )}$

• In a linear function space, i.e. a vector space consisting of functions from a set into a vector space , the null vector is the null function , where the null vector is the target space . ${\displaystyle A}$ ${\displaystyle W}$ ${\displaystyle f\equiv 0_{W}}$ ${\displaystyle 0_{W}}$

properties

Uniqueness

The zero vector of a vector space is unique. If there were two different zero vectors and , then applies immediately ${\displaystyle 0}$ ${\displaystyle {\bar {0}}}$

${\displaystyle 0=0+{\bar {0}}={\bar {0}}}$

and thus equality of the two vectors.

Scalar multiplication

The following applies to all scalars from the scalar body ${\displaystyle \alpha \in K}$

${\displaystyle \alpha \cdot 0_{V}=0_{V}}$

and analogously for all vectors of the vector space ${\displaystyle v\in V}$

${\displaystyle 0_{K}\cdot v=0_{V}}$,

which follows directly from the two distributive laws in vector spaces by choosing or . Together with it applies ${\displaystyle \alpha =\beta =0_{K}}$ ${\displaystyle u=v=0_{V}}$

${\displaystyle \alpha \cdot v=0_{V}\Leftrightarrow \alpha =0_{K}}$or , ${\displaystyle v=0_{V}}$

because it follows either or and then . ${\displaystyle \alpha \cdot v=0_{V}}$ ${\displaystyle \alpha =0_{K}}$ ${\displaystyle \alpha \neq 0_{K}}$ ${\displaystyle v=\alpha ^{-1}\cdot 0_{V}=0_{V}}$

Special rooms

In a scalar product , so a vector space with a scalar product , the zero vector is orthogonal to all vectors of the space, that is, for all vectors applies ${\displaystyle v\in V}$

${\displaystyle \langle 0_{V},v\rangle =\langle v,0_{V}\rangle =0_{K}}$,

what follows from the linearity or semi- linearity of the scalar product. In particular, the zero vector is thus also orthogonal to itself. In a normalized vector space , the norm of the zero vector applies

${\displaystyle \|0_{V}\|=0_{\mathbb {R} }}$

and the zero vector is the only vector with this property, which follows from the definiteness and the absolute homogeneity of the norm.

In a semi-normalized space there can be more than one vector whose norm is zero and such a vector is sometimes also called a zero vector. In a Minkowski space , light-like vectors are also referred to as zero vectors. In these cases, however, the concept of the zero vector does not correspond to the definition above.

Cross product

In three-dimensional Euclidean space , the cross product of any vector with the zero vector results in the zero vector again, that is ${\displaystyle V=\mathbb {R} ^{3}}$ ${\displaystyle 0\in \mathbb {R} ^{3}}$

${\displaystyle v\times 0=0\times v=0}$.

The same applies to the cross product of a vector with itself,

${\displaystyle v\times v=0}$.

Furthermore, the Jacobi identity applies , i.e. the cyclic sum of repeated cross products also gives the zero vector:

${\displaystyle u\times (v\times w)+v\times (w\times u)+w\times (u\times v)=0}$.

use

Linear combinations

For a given family of vectors with an index set , the zero vector can always be a linear combination ${\displaystyle (v_{i})_{i\in I}}$ ${\displaystyle I}$

${\displaystyle 0_{V}=\sum _{i\in I}\alpha _{i}\cdot v_{i}}$

express. The vectors are linearly independent if and only if all coefficients have to be in this linear combination . The zero vector can therefore never be part of a basis of a vector space, because it is already linearly dependent in itself. Each sub-vector space of a vector space contains at least the zero vector. The set , which consists only of the zero vector, forms the smallest possible sub-vector space of a vector space, the zero vector space ; its basis is the empty set , because the empty sum of vectors gives, by definition, the zero vector, i.e. ${\displaystyle \alpha _{i}=0_{K}}$ ${\displaystyle \{0_{V}\}}$ ${\displaystyle \emptyset }$

${\displaystyle \sum _{i\in \emptyset }v_{i}=0_{V}}$.

Linear maps

A linear mapping between two vector spaces and over the same scalar body always maps the zero vector onto the zero vector, because it applies ${\displaystyle T\colon V\to W}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle K}$

${\displaystyle T(0_{V})=T(0_{K}\cdot 0_{V})=0_{K}\cdot T(0_{V})=0_{W}}$.

However, further vectors from can also be mapped onto the zero vector of the target area. This set is called the kernel of the linear mapping and it forms a subspace of . A linear mapping is injective if and only if the kernel consists only of the zero vector. ${\displaystyle W}$ ${\displaystyle V}$ ${\displaystyle V}$

Linear equations

A homogeneous linear equation

${\displaystyle T(v)=0_{W}}$

therefore has at least the zero vector as a solution. It can be solved uniquely if and only if the kernel of the linear operator consists only of the zero vector. The reverse is an inhomogeneous linear equation ${\displaystyle v=0_{V}}$ ${\displaystyle T}$

${\displaystyle T(v)=w}$

with never solved by the zero vector. An inhomogeneous linear equation can be solved uniquely if the corresponding homogeneous equation only has the zero vector as a solution, which is a consequence of the superposition property . ${\displaystyle w\neq 0_{W}}$

literature

• Gilbert Strang : Linear Algebra . Springer, Berlin et al. 2003, ISBN 3-540-43949-8 .

Web links

• Eric W. Weisstein : Zero vector . In: MathWorld (English).