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
holds for all vectors . It is therefore the neutral element with regard to vector addition .
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 .
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.
- In the vector space of complex numbers , the zero vector is the number and thus also corresponds to the scalar zero.
- In the coordinate space , the zero vector is the n -tuple consisting of the zero elements of the body .
- In matrix space , the zero vector is the zero matrix , the elements of which are all equal .
- In the sequence space the zero vector is the sequence and not to be confused with the concept of the zero sequence .
- 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 .
properties
Uniqueness
The zero vector of a vector space is unique. If there were two different zero vectors and , then applies immediately
and thus equality of the two vectors.
Scalar multiplication
The following applies to all scalars from the scalar body
and analogously for all vectors of the vector space
- ,
which follows directly from the two distributive laws in vector spaces by choosing or . Together with it applies
- or ,
because it follows either or and then .
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
- ,
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
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
- .
The same applies to the cross product of a vector with itself,
- .
Furthermore, the Jacobi identity applies , i.e. the cyclic sum of repeated cross products also gives the zero vector:
- .
use
Linear combinations
For a given family of vectors with an index set , the zero vector can always be a linear combination
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.
- .
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
- .
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.
Linear equations
A homogeneous linear equation
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
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 .
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).