Null vector

The zero vector is in mathematics, a special vector of a vector space, namely the uniquely determined neutral element with respect to the vector addition. Examples of vectors are zero, the number is zero, the zero matrix and the zero function. A scalar product of the zero vector is orthogonal to all vectors of the space. In a normed space he is the only vector with zero norm. Every subspace of a vector space contains at least the zero vector, the smallest subspace of the vector space is zero. The zero vector is used to define some of the key concepts of linear algebra such as linear independence, basis and core. It plays an important role in the solution structure of linear equations.

• 5.1 linear combinations
• 5.2 Linear maps
• 5.3 Linear Equations

Definition

The zero vector of a vector space is the uniquely determined vector for which

Applies to all vectors. He is therefore the neutral element with respect to the vector addition.

Notation

The zero vector is usually by means of the digit zero, or just referred to. However, the zero vector is in general different from the zero element of Skalarkörpers the vector space, which is also represented by. If likelihood of confusion and therefore is the zero vector and denotes the scalar zero with. Occasionally, the zero vector is also supported by, or listed as a small o.

As the only vector of the Euclidean plane is the zero vector can not be represented graphically by an arrow, as it can be assigned neither direction nor length.

Examples

• In the vector space of real numbers is the zero vector, the number and thus equal to the zero of the Skalarkörpers.

• In the vector space of complex numbers is the zero vector, the number and thus also corresponds to the scalar zero.

• In the coordinate space is the zero vector, the n- tuple consisting of the zero elements of the body.

• In the die space of the zero vector is the zero matrix whose elements are all equal.

• In the next room is the zero vector is the consequence and not to be confused with the concept of zero sequence.

• In a linear function space, ie a vector space which consists of functions from a set into a vector space, the zero vector is the zero function, the zero vector of the target space is.

Properties

Unambiguity

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

For all scalars from the Skalarkörper

And similarly for all vectors of the vector space

Which follows directly from the two distributive in vector spaces by selecting or. Together applies so

As follows either or, then.

Special rooms

A scalar product, in other words a vector space with an inner product, the zero vector is orthogonal to all vectors of the space, that is, for all of the vectors is valid

Which follows from the linearity or semi -linearity of the scalar product. In particular, the zero vector is thus orthogonal to itself. In a normed vector space is valid for the norm of the zero vector

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

Cross product

In three-dimensional Euclidean space, the cross product of any vector with the zero vector yields again the zero vector, ie

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

Furthermore, the Jacobi identity applies, that is, the sum of repeated cyclic cross products also results in the null vector:

Use

Linear combinations

For a given family of vectors with an index set to the zero vector can always as a linear combination

Express. The vectors are linearly independent if in this linear combination must all coefficients. The zero vector can therefore never be part of a basis of a vector space, because it is already in itself linearly dependent. Every subspace of a vector space contains at least the zero vector. The crowd, which consists only of the zero vector, forms the smallest possible subspace of a vector space, the zero vector space; its base is the empty set, as the sum of empty vectors gives the zero vector, by definition, that is

Linear maps

A linear mapping between two vector spaces and over the same Skalarkörper always maps the zero vector to the zero vector, because it is

The zero vector of the target space but also more vectors can be mapped. This quantity is called the core of the linear mapping and it forms a subspace of. A linear map is injective if the core consists only of the zero vector.

Linear Equations

A homogeneous linear equation

Therefore has at least the zero vector as a solution. It is precisely then uniquely solvable if the core of the linear operator consists only of the zero vector. Conversely, a non-homogeneous linear equation

With never solved by the zero vector. An inhomogeneous linear equation is uniquely solvable if and only if the associated homogeneous equation has only the zero vector as a solution, which is a consequence of the superposition property.