Linear operator

Let $X$ and $Y$ be vector spaces over the same field $K$.

A function $T:X\to Y$ is a linear operator (or linear transformation) if:

1. $T(x+u)=T(x)+T(u)$ for all $x,u\in X$,
2. $T(\alpha x)=\alpha T(x)$ for all $\alpha\in K$ and $x\in X$.

Equivalently, $T$ is linear if and only if

One often writes $Tx$ instead of $T(x)$.

Linear operators are the morphisms of vector spaces . Their kernels and images give canonical subspaces, and the associated quotient describes the operator up to isomorphism.

Examples:

• Matrix multiplication $T(x)=Ax$ on $K^n$.
• The derivative $D:C^1[a,b]\to C[a,b]$, $Df=f'$ (linear over $\mathbb{R}$).
• The evaluation map $\mathrm{ev}_{x_0}:F(\Omega)\to K$, $\mathrm{ev}_{x_0}(f)=f(x_0)$.