# Linear mapping

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

## 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 α in K, the following two conditions are satisfied:

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

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.

## References

1. Lang, Serge (1987). Linear algebra. New York: Springer-Verlag. p. 51. .
2. Lax, Peter (2007). Linear Algebra and Its Applications, 2nd ed.. Wiley. p. 19. . (English)
3. Tanton, James (2005). Encyclopedia of Mathematics, Linear Transformation. Facts on File, New York. p. 316. . (English)
4. Sloughter, Dan (2001). "The Calculus of Functions of Several Variables, Linear and Affine Functions" (in English). Retrieved February 2014.