Theorem of Cayley-Hamilton

Let \(V\) be a vector space over the field \(K\) and \(f\in\mathrm{Hom}_K(V,V)\). If we define \(f\lambda := \lambda f\) for every \(\lambda\in K\) then we can substitute the linear map in the characteristic polynomial \(\chi_f\) (ring homomorphism). Further the linear map \[\phi:\mathrm{Hom}_K(V,V)\rightarrow K^{n\times n},f\mapsto M^E_E(f)\] is isomorph, where \(M^E_E(f)=:M\) is the image matrix of \(f\) regarding the canonical basis \(E\) of \(V\). We define again \(M\lambda := \lambda M\forall \lambda\in K\) and substitute in \(\chi_f\). \[\chi_f(\phi(f))=\mathrm{det}(M-\phi(f)I)=\mathrm{det}(M-M)=0\in K^{n\times n}\] Also note that \(\phi(f)^n=M^n=\phi(f^n)\) which yields for our polynomial where the \(\lambda_i\in K\) are coefficitients \[\chi_f(\phi(f))=\sum_{i=0}^n\lambda_i\phi(f)^i=\phi(\sum_{i=0}^n\lambda_i f^i)= \phi(\chi_f(f))\] Since \(\phi\) is isomorph we get \(\phi^{-1}(0)=0\in\mathrm{Hom}_K(V,V)\) and therefore \[\chi_f(f)=\chi_f(\phi^{-1}(\phi(f)))=\phi^{-1}\chi_f(\phi(f))=\phi^{-1}(0)=0\]

Every linear map is a root of its characteristic polynomial.