Linear map

In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.

The term linear transformation is especially used for linear maps from a vector space to itself (endomorphisms).

In abstract algebra, a linear map is a homomorphism of vector spaces.

Definition

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

$f({\mathbf {x}}+{\mathbf {y}})=f({\mathbf {x}})+f({\mathbf {y}})$ - additivity,

and

$f(a{\mathbf {x}})=af({\mathbf {x}})$ - homogenity.

This is equivalent to requiring that for any vectors x₁, ..., x_m and scalars a₁, ..., a_m, the equality

$f(a_{1}{\mathbf {x}}_{1}+\cdots +a_{m}{\mathbf {x}}_{m})=a_{1}f({\mathbf {x}}_{1})+\cdots +a_{m}f({\mathbf {x}}_{m})$

holds.