We introduced generalized eigenvectors and then we were given this example:
Given an endomorphism $\varphi: V \to V$ with $\chi_\varphi(t) = (-1)^n (t-\lambda)^n$, then $U(\lambda) = V$ since $(\varphi-\lambda · \mathrm{id})^n = 0$.
I don't really get this. Why is it that $(\varphi-\lambda · \mathrm{id})^n = 0$? If this is the case then clearly $U(\lambda) = V$ but I can't justify this.