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.

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:

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

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

Sometimes, a linear mapping is called a linear function. However, in basic mathematics, a linear function means a function whose graph is a line (i.e. ${\displaystyle f(x)=ax+b}$). The set of all linear mappings from the vector space V to vector space W can be written as ${\displaystyle L(V,W)}$.

Related pages

• Homomorphism
• Transformation

References

The English Wikibooks has more information on:

Linear Algebra/Linear Transformations

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