Referenzbasis

Referenzbasis is a term used in mathematics, physics and related disciplines to denote a chosen basis of a vector space that serves as a standard or reference for expressing objects, comparing states and performing coordinate transformations.

In linear algebra, a basis B = {b1, …, bn} of a vector space V is a set of linearly independent vectors that span V. When B is fixed, every vector v ∈ V has unique coordinates (c1, …, cn) such that v = c1 b1 + … + cn bn. The designation “Referenzbasis” emphasizes that this basis is selected as a reference frame for representation, measurement, or computation.

In physics and quantum mechanics, a referenzbasis often corresponds to a measurement basis or a preferred set of eigenstates, such as an energy eigenbasis or a spin basis. Coefficients of a state in that basis give the probabilities or amplitudes for observing corresponding outcomes.

In signal processing, data science and numerical analysis, a referenzbasis can be the standard basis in R^n or a chosen dictionary used for sparse representations. Transforming coordinates between bases is accomplished via a basis transformation matrix, with v = ∑ ci bi and the coordinate vector c = [c1, …, cn]^T.

The choice of Referenzbasis influences computational ease, interpretability, and numerical properties. Orthonormal bases (for example Fourier or wavelet bases) are popular because they simplify projections and preserve energy.

Note that the term is context dependent; in some texts it is used interchangeably with Basis, Standardbasis or Referenzsystem. The core idea remains: a referenzbasis provides a fixed, standard reference for expressing and comparing elements of a vector space.