= Solution
If $\dim V=n$, then $\dim\operatorname{End}(V)=n^2$. The $n^2+1$ <endomorphisms> $I,T,\ldots,T^{n^2}$ are therefore <linearly dependent>, so some nonzero <polynomial> $p$ satisfies $p(T)=0$. The <minimal polynomial> is the <monic polynomial> of least degree annihilating $T$. Uniqueness follows because the difference of two such <monic polynomials> would be a nonzero annihilator of smaller degree.
A <Jordan block> $J_r(\lambda)$ has minimal polynomial $(x-\lambda)^r$, and the minimal polynomial of a direct sum is the <least common multiple> of those of its blocks. Thus an eight-dimensional example is
$$
\boxed{T=J_2(0)\oplus J_3(1)\oplus J_3(1),\qquad m_T(x)=x^2(x-1)^3.}
$$
Back to article page
