Eigentheory/R/One-dimensional/Elementary/Example

A linear mapping from ${}\mathbb {R}$ to ${}\mathbb {R}$ is the multiplication with a fixed number ${}a\in \mathbb {R}$ (the proportionality factor). Therefore, every number ${}v\neq 0$ is an eigenvector for the eigenvalue ${}a$, and the eigenspace for this eigenvalue is the whole ${}\mathbb {R}$. Beside ${}a$, there are no other eigenvalues, and all eigenspaces for ${}\lambda \neq a$ are ${}0$.