= Solution
The <minimal polynomial> $m_T(t)$ is the unique monic <polynomial> of least positive <polynomial degree> satisfying $m_T(T)=0$. Such an annihilating <polynomial> exists without using the <Cayley-Hamilton theorem>: the <vector space> $\operatorname{End}(V)$ has <dimension of a vector space> equal to $n^2$, so $I,T,\ldots,T^{n^2}$ are <linearly dependent>. Normalize a nonzero annihilating <polynomial> to be monic and choose one of least degree. Two distinct monic choices of that degree would have an annihilating difference of lower degree, contradicting minimality. A nonzero constant cannot annihilate $T$ because $V\ne0$.
Choose a <basis> $B$ and let $A=[T]_B$. The <characteristic polynomial> is
$$
\boxed{\chi_T(t)=\det(tI-A).}
$$
It is independent of the chosen <basis>. Indeed a change of basis replaces $A$ by a <similar matrix> $P^{-1}AP$, and
$$
\det(tI-P^{-1}AP)=\det\bigl(P^{-1}(tI-A)P\bigr)=\det(tI-A)
$$
by multiplicativity of the <determinant>. Thus this monic degree-$n$ <polynomial> is an invariant of the <linear transformation> itself.
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