Linear map

A linear map is a function $T:V\to W$ between vector spaces over the same field $\mathbb{F}$ such that for all $u,v\in V$ and $a\in\mathbb{F}$,

As a function, $T$ has domain $V$ and codomain $W$. The special case $V=W$ is a linear operator .

Examples:

• For a fixed matrix $A$, the map $T(x)=Ax$ from $\mathbb{F}^n$ to $\mathbb{F}^m$ is linear.
• The derivative map $D:\mathbb{R}[x]\to\mathbb{R}[x]$ given by $D(p)=p'$ is linear.
• The projection $P:\mathbb{R}^2\to\mathbb{R}^2$ defined by $P(x,y)=(x,0)$ is linear.