# Linear map

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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 : VW 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:

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

and

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

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

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

holds.