Linear map

Let $V$ and $W$ be vector spaces over the same field $\mathbb{F}$ (typically $\mathbb{F}=\mathbb{R}$ or $\mathbb{C}$). A function $T:V\to W$ is a linear map (or linear transformation) if for all $x,y\in V$ and all scalars $\alpha,\beta\in\mathbb{F}$,

Linear maps are the morphisms in linear algebra; in analysis they model derivatives (as best linear approximations) and bounded linear operators (when norms are present). See also operator norm .

Examples:

• If $A$ is an $m\times n$ real matrix, then $T:\mathbb{R}^n\to\mathbb{R}^m$ defined by $T(x)=Ax$ is linear.
• The derivative operator $D:C^1([a,b])\to C([a,b])$, $D(f)=f'$, is linear.
• The map $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=x^2$ is not linear.