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.