Let be a matrix over a field . Prove that the dimension of the minimal polynomial of is at most .
If , note that the set is isomorphic to once we select a pair of bases. The set of all matrices over is naturally an -vector space of dimension . Consider now the powers of : , , , …, . These matrices, as elements of , are necessarily linearly dependent, so that for some . The minimal polynomial divides this , and so has degree at most .