A *linear transformation*, $\ltdefn{T}{U}{V}$, is a function that carries elements of the vector space $U$ (called the *domain*) to the vector space $V$ (called the *codomain*), and which has two additional properties

- $\lt{T}{\vect{u}_1+\vect{u}_2}=\lt{T}{\vect{u}_1}+\lt{T}{\vect{u}_2}$ for all $\vect{u}_1,\,\vect{u}_2\in U$
- $\lt{T}{\alpha\vect{u}}=\alpha\lt{T}{\vect{u}}$ for all $\vect{u}\in U$ and all $\alpha\in\complex{\null}$