$A$ be $n×n$ matrix $A^{n}=0$ ,$A^{n-1}$ not equal to zero a vector $v$ belongs to R^n.then how to proof {V,AV,...A^(n-1)V} is a basis.

Question

Given $A$ be $n×n$ matrix such that $A^{n}=0$, but $A^{n-1}$ not equal to zero a vector $v$ belongs to $\Bbb{R}^{n}$. Proof that {$V,AV,\cdots,A^{(n-1)}V$} is a basis.

Answer 1

Is enough prove that $\{v,Av,...,A^{n-1}v\}$ is LI. If $a_{1},...,a_{n}$ is such that $$a_{1}v+...+a_{n}A^{n-1}v=0 ,$$ applying $A^{n-1}$ in this equation, we have $$a_{1}A^{n-1}v=0 ,$$ so $a_{1}=0$. Similarly one can prove that $a_{2}=...=a_{n}=0$.