# Linear mapping

mapping that preserves the operations of addition and scalar multiplication

In mathematics (particularly in linear algebra), a linear mapping (or linear transformation) is a mapping f between vector spaces that preserves addition and scalar multiplication.[1][2][3]

Mirroring as long an axis is an example of a linear mapping

## Definition

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

 ${\displaystyle f(\mathbf {x} +\mathbf {y} )=f(\mathbf {x} )+f(\mathbf {y} )\!}$ ${\displaystyle f(\alpha \mathbf {x} )=\alpha f(\mathbf {x} )\!}$

Sometimes, a linear mapping is called a linear function.[4] However, in basic mathematics, a linear function means a function whose graph is a line. The set of all linear mappings from the vector space V to vector space W can be written as ${\displaystyle L(V,W)}$ .[5]

## References

1. Lang, Serge (1987). Linear algebra. New York: Springer-Verlag. p. 51. ISBN 9780387964126.
2. Lax, Peter (2007). Linear Algebra and Its Applications, 2nd ed. Wiley. p. 19. ISBN 978-0-471-75156-4. (in English)
3. Tanton, James (2005). Encyclopedia of Mathematics, Linear Transformation. Facts on File, New York. p. 316. ISBN 0-8160-5124-0. (in English)
4. Sloughter, Dan (2001). "The Calculus of Functions of Several Variables, Linear and Affine Functions" (PDF). Retrieved 1 February 2014.
5. "Comprehensive List of Algebra Symbols". Math Vault. 2020-03-25. Retrieved 2020-10-12.