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Vector space, direct sums involving linear maps and polynomials

Let be $V$ a vector space over $K$ and lets $a\in K\setminus\{0\}$ and $T:V \to V$ a linear map s. t. $a^2T-3aT^2+T^3=0$. Show that $V = Ker(T) \oplus Im(T)$. My professor told me an observation: This result is more general, that is, if I have a polynomial $P$ such with $P(0) = 0$ and $P'(0) […]