Matrix/K/Linear dependence of powers/Exercise

Let ${}K$ be a field, and let ${}M$ denote an ${}n\times n$-matrix over ${}K$. Show that the first ${}n^{2}+1$ powers

$M^{i},i=0,\ldots ,n^{2},$

are linearly independent in ${}\operatorname {Mat} _{n}(K)$.

Create a solution