ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Map
Operation
N-operation
Binary operation
Enclosed binary operation
Groupoid
Ringoid
Semiring
Ring
Left ring action
Module
Linear combination
Definition D690
Linear map
Formulation 1
Let $R$ be a D273: Division ring such that
(i) $V$ and $W$ are each a D29: Vector space over $R$
(ii) $f : V \to W$ is a D18: Map from $V$ to $W$
Then $f$ is a linear map from $V$ to $W$ over $R$ if and only if \begin{equation} \forall \, N \in \{ 1, 2, 3, \ldots \} : \forall \, x \in V^N : \forall \, r \in R^N : f \left( \sum_{n = 1}^N r_n x_n \right) = \sum_{n = 1}^N r_n f(x_n) \end{equation}
Children
Affine map
Conic map
Sublinear map
Superlinear map
Vector space isomorphism