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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.
Sammon mapping; Scalar field; Second derivative; Self-concordant function; Semi-differentiability; Semilinear map; Set function; List of set identities and relations; Shear mapping; Shekel function; Signomial; Similarity invariance; Soboleva modified hyperbolic tangent; Softmax function; Softplus; Splitting lemma (functions) Squeeze theorem ...
In functional analysis and related areas of mathematics, an almost open map between topological spaces is a map that satisfies a condition similar to, but weaker than, the condition of being an open map. As described below, for certain broad categories of topological vector spaces, all surjective linear operators are necessarily almost open.
For instance, a "map" is a "continuous function" in topology, a "linear transformation" in linear algebra, etc. Some authors, such as Serge Lang , [ 8 ] use "function" only to refer to maps in which the codomain is a set of numbers (i.e. a subset of R or C ), and reserve the term mapping for more general functions.
A squeeze mapping moves one purple hyperbolic sector to another with the same area. It also squeezes blue and green rectangles.. In 1688, long before abstract group theory, the squeeze mapping was described by Euclid Speidell in the terms of the day: "From a Square and an infinite company of Oblongs on a Superficies, each Equal to that square, how a curve is begotten which shall have the same ...
In mathematics, an invariant subspace of a linear mapping T : V → V i.e. from some vector space V to itself, is a subspace W of V that is preserved by T. More generally, an invariant subspace for a collection of linear mappings is a subspace preserved by each mapping individually.
The assignment produces an injective linear map between the space of linear operators from to and the space of linear operators from # to #. If X = Y {\displaystyle X=Y} then the space of linear maps is an algebra under composition of maps , and the assignment is then an antihomomorphism of algebras, meaning that t ( u v ) = t v t u ...
Given a module M over a ring R, a linear form on M is a linear map from M to R, where the latter is considered as a module over itself. The space of linear forms is always denoted Hom k (V, k), whether k is a field or not. It is a right module if V is a left module.