ithkomponenten

The ithkomponenten, or i-th component, denotes the element in position i of a finite sequence, vector, or array. For a vector x = (x_1, x_2, ..., x_n) in R^n, the i-th component is denoted x_i and is the coordinate along the i-th dimension of the chosen coordinate system.

In mathematics the index i ranges from 1 to n. In computer science and programming, arrays are often zero-based, so the i-th component corresponds to x[i-1] in a 0-based indexing scheme.

The concept is used to describe coordinates of a point, coefficients in linear combinations, and components of functions φ_i: R^n -> R defined by φ_i(x) = x_i. This viewpoint emphasizes that a vector is a collection of its coordinates, each coordinate functioning as a separate scalar component.

Extensions to tensors and multi-dimensional data: the notion generalizes to higher-order arrays; the i-th component of a tensor is obtained by fixing all but one index. In data analysis, the i-th component can refer to the i-th feature across samples; in physics, coordinates of a position vector are the i-th components along the chosen basis.

See also: vector component, coordinate, basis, Cartesian coordinates, indexing.