Linear Map

Let $V$ and $W$ be vector spaces over the same field $F$. A linear map (or linear transformation) is a function $T:V\to W$ such that for all $u,v\in V$ and all $a\in F$,

Equivalently, $T$ preserves all finite linear combinations: for any $v_1,\dots,v_n\in V$ and $a_1,\dots,a_n\in F$,

Given such a $T$, its kernel (or null space) is $\ker(T)=\{v\in V:T(v)=0\}$ and its image is $\operatorname{im}(T)=\{T(v):v\in V\}$. A subset $U\subseteq V$ is a (linear) subspace if $u,u'\in U$ and $a\in F$ implies $u+u'\in U$ and $a\cdot u\in U$; with this definition, both $\ker(T)$ and $\operatorname{im}(T)$ are subspaces.

In finite-dimensional situations, rank-nullity relates the “size” of $\ker(T)$ and $\operatorname{im}(T)$.

Examples:

• The coordinate projection $\pi:\mathbb{R}^3\to\mathbb{R}^2$, $\pi(x,y,z)=(x,y)$ (between instances of Euclidean space ), is linear.
• For a field $F$, the derivative map $D:F[t]\to F[t]$, $D(p)=p'$, is linear.
• Fix $t_0\in F$. The evaluation map $\operatorname{ev}_{t_0}:F[t]\to F$, $\operatorname{ev}_{t_0}(p)=p(t_0)$, is linear.