Linear Map

definitionvector-spaces

Definition

A linear map or linear transformation from $V$ to $W$ is a function $T:V\to W$ satisfying,

1. (Additivity) $T(u+v)=Tu+Tv\forall u,v\in V$.
2. (Homogeneity) $T(\lambda u)=\lambda (Tu)\forall \lambda \in \mathbb{F},u\in V$.

Linear Isomorphism

If a linear map $T$ is bijective, then it is a linear isomorphism.

Linear Operator

If $T:V\to W$ is linear, and $V=W$, then we call $T$ a linear operator.

Examples