Vektorrepresentation

Vektorrepresentation, or vector representation, is the expression of an object as an element of a vector space using a coordinate vector or a set of features. In mathematics, a vector is an ordered n-tuple that can be added to and scaled by scalars; the specific coordinates depend on the chosen basis for the space.

A vector can be written as v = sum_i c_i b_i, where B = {b1, ..., bn} is a basis and c_i are the coordinates. The same vector has different coordinates in different bases, related by a linear transformation represented by a change-of-basis matrix. This highlights that a vector’s numerical representation is relative to the chosen basis and scale.

In computer science and data analysis, objects are often represented as vectors of numbers, called feature vectors or embeddings. Common representations include dense vectors and sparse vectors (for example, one-hot encodings). Learned embeddings, such as word, sentence, or graph embeddings, capture semantic or structural information in a continuous vector space.

Vector representations enable standard operations: addition, scalar multiplication, inner products, norms, and distance measures, as well as the action of linear maps via matrices. Changing basis corresponds to multiplying by a matrix, and metric properties can depend on the chosen inner product.

Applications span similarity search, clustering, regression, and dimensionality reduction. The choice of representation influences algorithm performance and interpretability. For example, a point in three-dimensional space has coordinates (2, -1, 5) in the standard basis, but a different basis yields different coordinates that describe the same point. In machine learning, vector representations are fundamental for processing and learning from data.