Let A be an n x n complex Jordan Block. I am trying to show that there exists a vector $\vec{v}$ such that $\vec{v} , A \vec{v}, \dots , A^{n-1} \vec{v}$ constitute a basis for $\mathbb{C}^n$.
As I explore this question, I know it has a lot to do with Jordan Chains. However, is there a way to prove it without calling out a Jordan Chain?