0vektoren

A 0vektoren, also known as the zero vector, is a fundamental concept in linear algebra and vector spaces. It is a vector with a magnitude of zero, meaning it has no length or direction. The zero vector is denoted by various symbols, including **0**, **0⃗**, or simply **⃗0**, depending on the context.

In an n-dimensional space, the zero vector consists of n components, each of which is zero. For example, in three-dimensional space, the zero vector is represented as (0, 0, 0). It serves as the additive identity in vector addition, meaning that when it is added to any other vector, the result is the original vector. Mathematically, for any vector **v**, the following holds:

**v + 0 = v**

The zero vector is also the multiplicative identity in scalar multiplication, as multiplying any vector by a scalar of zero yields the zero vector. This property is expressed as:

**0 · v = 0**

The zero vector plays a crucial role in defining vector spaces, where it is one of the required axioms. It ensures that every vector in the space has a unique additive inverse, which, when added to the original vector, results in the zero vector. The zero vector is also essential in discussions of linear independence, spanning sets, and the rank of matrices.

In practical applications, the zero vector often represents a state of equilibrium or no change, such as zero displacement in physics or zero error in computational models. Its simplicity and universal properties make it a cornerstone of vector-based mathematical frameworks.