Kdimensionalen

Kdimensionalen refers to the characteristic of having k independent directions or degrees of freedom, where k is a nonnegative integer. In mathematics and related fields, k-dimensional objects are modeled in k-dimensional spaces such as Euclidean k-space, denoted R^k, where a point is specified by an ordered k-tuple (x1,...,xk).

Distances are defined by the Euclidean metric: d(x,y) = sqrt(sum (xi-yi)^2). The dimension k equals the size of a basis of the underlying vector space, meaning every point can be expressed as a linear combination of k basis vectors.

Subspaces with lower dimension exist, such as hyperplanes of dimension k-1, and standard examples include the 2-dimensional plane and the 3-dimensional space.

In applied contexts, high-dimensional spaces are used to model data with k features. Techniques such as dimensionality reduction (PCA, t-SNE) aim to project data into fewer than k dimensions while preserving structure. The term also appears in geometry, where k-dimensional polytopes, spheres S^{k-1}, and other figures generalize from familiar 2D and 3D cases.

Beyond finite k, there are infinite-dimensional spaces studied in functional analysis, but k-dimensional notions are typically confined to finite k.

Related concepts include dimension, Cartesian space, Euclidean space, manifolds, and dimensionality reduction.