In mathematics, a linear map, linear mapping, linear transformation, or linear operator (in some contexts also called linear function) is a function between two modules (including vector spaces) that preserves the operations of module (or vector) addition and scalar multiplication.
As a result, it always maps straight lines to straight lines or to a single point. The expression "linear operator" is commonly used for linear maps from a vector space to itself (i.e., endomorphisms). Sometimes the definition of a linear function coincides with that of a linear map, while in analytic geometry it does not.
In the language of abstract algebra, a linear map is a homomorphism of modules. In the language of category theory it is a morphism in the category of modules over a given ring.
Read more about Linear Map: Definition and First Consequences, Examples, Matrices, Examples of Linear Transformation Matrices, Forming New Linear Maps From Given Ones, Endomorphisms and Automorphisms, Kernel, Image and The Rank–nullity Theorem, Cokernel, Algebraic Classifications of Linear Transformations, Change of Basis, Continuity, Applications
Famous quotes containing the word map:
“When I had mapped the pond ... I laid a rule on the map lengthwise, and then breadthwise, and found, to my surprise, that the line of greatest length intersected the line of greatest breadth exactly at the point of greatest depth.”
—Henry David Thoreau (1817–1862)