Transform

Transform is a general mathematical concept referring to a correspondence that maps elements of one set to elements of another set, often preserving some structure. In mathematics and applied fields, transforms are used to reinterpret objects in a form that reveals properties or simplifies computation. They can be continuous (functions) or discrete (finite sequences).

Linear and affine transforms: A linear transformation maps vectors to vectors and can be represented by a matrix. Affine transforms include linear transformation followed by a translation. Geometric transforms such as rotation, scaling, and reflection belong to this family. Invertibility depends on the underlying matrix or transformation.

Coordinate and geometric transforms: Changing coordinate systems (basis change) reexpresses objects in different coordinates. Geometric transforms in graphics manipulate position, size, or orientation; homogeneous coordinates enable projective transforms.

Common analytical transforms: Fourier transform decomposes a time-domain signal into frequency components; Laplace transform converts functions of time into the complex frequency domain; Z-transform does the same for discrete-time sequences. Wavelet transforms and the Short-Time Fourier Transform offer time–frequency localization.

In computing and artificial intelligence, the term transform also appears in the Transformer model, a neural network architecture that uses attention mechanisms to transform input sequences into contextualized representations.