In other words, considering the matrix as one with coefficients in a larger field does not change the minimal polynomial.
The reason for this differs from the case with the characteristic polynomial (where it is immediate from the definition of determinants), namely by the fact that the minimal polynomial is determined by the relations of linear dependence between the powers of A: extending the base field will not introduce any new such relations (nor of course will it remove existing ones).
In particular one has: These cases can also be proved directly, but the minimal polynomial gives a unified perspective and proof.
For a nonzero vector v in V define: This definition satisfies the properties of a proper ideal.
and for these coefficients one has Define T to be the endomorphism of R3 with matrix, on the canonical basis, Taking the first canonical basis vector e1 and its repeated images by T one obtains of which the first three are easily seen to be linearly independent, and therefore span all of R3.