Teisendid

Teisendid are mappings between mathematical structures that apply a rule to each element of a set to produce an element in another set, often preserving some kind of structure. They are studied across fields such as geometry, algebra, and analysis, and they underpin many ways of transforming objects or data.

In linear algebra, a common type is the linear teisend T: V -> W between vector spaces. Such a teisend satisfies T(x + y) = T(x) + T(y) and T(a x) = a T(x) for all vectors x, y and scalars a. Linear teisendid can be represented by matrices with respect to chosen bases, and properties such as kernel, image, rank, and invertibility are central. Affine teisendid extend linear ones by including a translation term: x maps to A x + b.

Geometric teisendid include rotations, translations, reflections, scalings, and more general projective transformations. Some preserve distances or angles (isometries), while others preserve parallelism or incidence relations. In many contexts, a teisend is bijective, with an inverse that undoes the transformation.

In analysis and signal processing, transforms such as the Fourier teisend, Laplace teisend, and other integral transforms convert functions to different representations, often revealing frequency or temporal properties and enabling inversion to recover the original function.

Notation for a teisend is typically T: X -> Y or f: X -> Y, with some being linear, affine, or nonlinear. The composition of teisendid yields new transformations, and their study includes questions of existence, uniqueness, invariants, and fixed points.