The question is formatted as follows. Firstly I am given the following theorem:
If T is a linear operator, and A is its matrix representation on $\mathbb{C}^n$, and the minimum polynomial is $m_T(x)=p(x)q(x)$ where $p(x)$ and $q(x)$ are coprime polynomials, then $Im(q(T))=Ker(p(T))$
The the question continues as follows:
Let $p(x)=(x-\mu)^m$, and $x_1, x_2, ..., x_m$ be the set of generalized eigenvectors generated by $x_m$ of grade $m$. Show that $x_m, (A-\mu I)x_m, (A-\mu I)^2x_m,\dots, (A-\mu I)^{m-1}x_m$ is a basis for the generalized eigenspace.
I managed to show that they are linearly independent but did not manage to show how they are spanning. I tried using the first part of the question to no avail. Any hints?
My attempt so far is as follows. For each one of the vectors in the set, $p(A)(A-\mu I)^ix_m=0$ which implies that $(A-\mu I)^ix_m\in Ker(p(T))=Im(q(T))$. Thereofore $\forall i \exists v_i$ such that $q(A)v_i=x_i=(A-\mu I)^i x_m$.
But this led me nowhere