 Solution

The minimal polynomial of a linear transformation $T$ is the unique monic polynomial of least degree among the nonzero polynomials $p\in F[x]$ satisfying $p(T)=0$:Such a polynomial exists because the endomorphism space is finite-dimensional, so sufficiently many powers $I,T,T^2,\dots$ are linearly dependent. Monic normalization is possible over the field $F$. Two monic polynomials annihilating $T$ of the same least degree have a difference of smaller degree annihilating $T$, so their difference is zero. For the zero vector space the convention is $m_T=1$.