Past exam of the linear algebra course of the University of Oxford / 2017 / 1 / a / i

For a linear transformation $T$ of a finite-dimensional vector space over a field $F$, its minimal polynomial $m_T$ is the unique monic polynomial of least degree satisfying $m_T(T)=0$. There is at least one nonzero polynomial with this property because the powers $I,T,T^2,\dots$ lie in the finite-dimensional vector space of linear maps $V\to V$. Dividing any other such polynomial by $m_T$ leaves a remainder of smaller degree that also vanishes at $T$, so that remainder must be zero. This also proves uniqueness.

If $C$ represents $T$ in any basis, the paper's characteristic polynomial isIt is independent of the basis, since changing the basis replaces $C-xI$ by a similar matrix, preserving its determinant. Its degree is $\dim V$ and its leading coefficient is $(-1{)}^{\dim V}$; $m_T$ is always monic.