Linear mapping

Mirroring along an axis is an example of a linear mapping

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]

Definition

Let V and W be vector spaces over the same field K. A function f: V → W 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:

[math]\displaystyle{ f(\mathbf{x}+\mathbf{y}) = f(\mathbf{x})+f(\mathbf{y}) \! }[/math]

[math]\displaystyle{ f(\alpha \mathbf{x}) = \alpha f(\mathbf{x}) \! }[/math]

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 [math]\displaystyle{ L(V, W) }[/math].^[5]

Linear Mapping Media

The function f:\R^2 \to \R^2 with f(x, y) = (2x, y) is a linear map. This function scales the x component of a vector by the factor 2.
The function f(x, y) = (2x, y) is additive: It does not matter whether vectors are first added and then mapped or whether they are mapped and finally added: f(\mathbf a + \mathbf b) = f(\mathbf a) + f(\mathbf b)
The function f(x, y) = (2x, y) is homogeneous: It does not matter whether a vector is first scaled and then mapped or first mapped and then scaled: f(\lambda \mathbf a) = \lambda f(\mathbf a)