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In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication.
A constant function is also considered linear in this context, as it is a polynomial of degree zero or is the zero polynomial. Its graph, when there is only one variable, is a horizontal line. In this context, a function that is also a linear map (the other meaning) may be referred to as a homogeneous linear function or a linear form.
Linear fractional transformations leave cross ratio invariant, so any linear fractional transformation that leaves the unit disk or upper half-planes stable is an isometry of the hyperbolic plane metric space. Since Henri Poincaré explicated these models they have been named after him: the Poincaré disk model and the Poincaré half-plane model.
Affine transformation (Euclidean geometry) Bäcklund transform; Bilinear transform; Box–Muller transform; Burrows–Wheeler transform (data compression) Chirplet transform; Distance transform; Fractal transform; Gelfand transform; Hadamard transform; Hough transform (digital image processing) Inverse scattering transform; Legendre ...
Linear functionals first appeared in functional analysis, the study of vector spaces of functions. A typical example of a linear functional is integration: the linear transformation defined by the Riemann integral = is a linear functional from the vector space [,] of continuous functions on the interval [,] to the real numbers.
Normally, a matrix represents a linear map, and the product of a matrix and a column vector represents the function application of the corresponding linear map to the vector whose coordinates form the column vector. The change-of-basis formula is a specific case of this general principle, although this is not immediately clear from its ...
A composition of four mappings coded in SVG, which transforms a rectangular repetitive pattern into a rhombic pattern. The four transformations are linear.. In mathematics, a transformation, transform, or self-map [1] is a function f, usually with some geometrical underpinning, that maps a set X to itself, i.e. f: X → X.