forvandlinger

Forvandlinger, commonly translated as transformations, is a concept used across mathematics, science, and computing to describe an operation that converts an object into another form. In its most general sense, a transformation is a relation that assigns to every element of a domain a unique element of a codomain; that is, it is a function or mapping. When a transformation preserves certain structure, it is often studied for its symmetry or invariants.

In geometry, transformations include translations, rotations, reflections and scaling. More general families, such as affine transformations (combinations of linear transformations and translations) and projective transformations, act on spaces of points and preserve specific features, such as collinearity or ratios of distances along lines. Rigid motions, which preserve distances and angles, are a subclass of isometries.

In linear algebra, a linear transformation maps a vector space to itself or to another vector space and respects addition and scalar multiplication. They can be represented by matrices; properties such as kernel, image and determinant describe their size and invertibility. Composition of transformations corresponds to matrix multiplication and forms a group under invertible transformations.

In analysis, integral or Fourier transforms are transformations that map functions into other functions, often simplifying the handling of differential equations or signal processing. In physics and computer science, transformations underpin coordinate changes, symmetry operations, graphics pipelines, and robotics.

Historically, the formal study of transformations grew from geometry and linear algebra in the 19th and 20th centuries, with developments in group theory and linear mappings shaping modern mathematics. The concept also appears in applied disciplines under names such as coordinate transformations, similarity transforms, and transform methods.