If M^(n-1) notequal 0 but $\displaystyle , M^n=0 $, then determine dim ker (M)
I feel the following two equations and rank nullity theorem can be used for this but not sure how
Let $\displaystyle x$ such that $\displaystyle M^{n-1}x\neq 0$. Show that the vectors $\displaystyle x,Mx,\ldots,M^{n-1}x$ are linearly independent. Therefore, the family $\displaystyle \mathcal B=\{M^jx\}_{0\leq j\leq n-1}$ is a basis of the vector space in which we are working. Write the endomorphism associated to $\displaystyle M$ in the basis $\displaystyle \mathcal B$.