# Linear map

From Citizendium

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:

- - additivity,

*and*

- - homogenity.

This is equivalent to requiring that for any vectors **x**_{1}, ..., **x**_{m} and scalars *a*_{1}, ..., *a*_{m}, the equality

holds.