Linear operator

A linear operator is a linear map $T:V\to V$ on a vector space $V$.

Because the domain and codomain agree, one can form iterates $T^k$ using composition . Linear operators are the objects to which eigenvalues , eigenspaces , and the characteristic polynomial are attached.

Examples:

• The identity map $I:V\to V$ given by $I(v)=v$ is a linear operator.
• On $\mathbb{R}^2$, rotation by a fixed angle is a linear operator.
• On $\mathbb{R}[x]$, the differentiation map $p\mapsto p'$ is a linear operator.