Linear map

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In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.

The term linear transformation is especially used for linear maps from a vector space to itself (endomorphisms).

In abstract algebra, a linear map is a homomorphism of vector spaces.

Definition

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

f(\bold{x}+\bold{y})=f(\bold{x})+f(\bold{y}) - additivity,

and

f(a \bold{x})=a f(\bold{x}) - homogenity.

This is equivalent to requiring that for any vectors x1, ..., xm and scalars a1, ..., am, the equality

f(a_1 \bold{x}_1+\cdots+a_m \bold{x}_m)=a_1 f(\bold{x}_1)+\cdots+a_m f(\bold{x}_m)

holds.

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