Linear Operator

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

Linear operators are the basic input for spectral notions such as eigenvalues and eigenvectors, and for polynomial invariants like characteristic and minimal polynomials.

Examples:

• The identity operator $I:V\to V$ given by $I(v)=v$ is linear.
• The zero operator $0:V\to V$ given by $0(v)=0$ is linear.
• On $\mathbb{R}^2$, the map $T(x,y)=(x,0)$ is a linear operator (projection onto the $x$-axis).