Linear map

related topics
{math, number, function}
{disease, patient, cell}

In mathematics, a linear map, linear transformation, or linear operator (in some contexts also called linear function) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. The expression "linear operator" is commonly used for linear maps from a vector space to itself (endomorphisms). In advanced mathematics, the definition of linear function coincides with that of linear map, while in analytic geometry it is less strict.

In the language of abstract algebra, a linear map is a homomorphism of vector spaces, or in the language of category theory a morphism in K-Vect, the category of vector spaces over a given field K.

Contents

Definition and first consequences

Let V and W be vector spaces over the same field K. A function f : VW is said to be a linear map if for any two vectors x and y in V and any scalar α in K, the following two conditions are satisfied:

This is equivalent to requiring that for any vectors x1, ..., xmV and scalars a1, ..., amK, the following equality holds:

It immediately follows from the definition that f(0) = 0.

Occasionally, V and W can be considered to be vector spaces over different fields. It is then necessary to specify which of these ground fields is being used in the definition of "linear". If V and W are considered as spaces over the field K as above, we talk about K-linear maps. For example, the conjugation of complex numbers is an R-linear map CC, but it is not C-linear.

Full article ▸

related documents
Universal property
Euler–Mascheroni constant
Horner scheme
Axiom schema of specification
NP-complete
Convergence of random variables
Splay tree
Preadditive category
Field extension
Bubble sort
Spectral theorem
Euler–Maclaurin formula
Self-organizing map
Hausdorff space
Binary tree
Additive category
Binary heap
Planar graph
Projective plane
Polymorphism in object-oriented programming
Brute-force search
Rice's theorem
Transcendental number
Logical connective
Inequality
Constructivism (mathematics)
Partial derivative
Isomorphism
Total order
Analytic continuation