# 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.