dimensionpreserving

Dimensionpreserving, usually written dimension-preserving, is a descriptive term used in mathematics and related fields to denote a property of a map, transformation, or process that does not change the dimension of the objects involved. The precise meaning depends on context, as dimension can refer to vector-space dimension, topological dimension, or other notions such as Hausdorff dimension.

In linear algebra, dimension refers to the dimension of vector spaces. A linear map T: V → W between finite-dimensional spaces preserves dimension on a subspace U ⊆ V if dim T(U) = dim U. This holds exactly when the restriction T|U is injective. Global dimension preservation occurs when T is an isomorphism, in which case dim V = dim W and the dimensions of all subspaces are preserved under T.

In topology and dimension theory, a map f: X → Y is dimension-preserving if dim f(X) = dim X, where dim denotes the chosen notion of dimension (topological, covering, or Hausdorff dimension). Many continuous maps do not preserve dimension; for example, certain projections can reduce dimension. Some classes of maps, such as homeomorphisms between spaces of the same dimension or dimension-preserving embeddings, guarantee preservation.

In applications, dimension-preserving transformations are common in geometry and computer graphics, where rigid motions (translations, rotations, reflections) preserve ambient dimension. In data analysis, dimension-preserving embeddings aim to keep the intrinsic dimensionality of the data intact, while dimension-reducing methods (projections, many nonlinear embeddings) typically alter dimension.