Chapter 3.3, Problem 78E

An $n\times n$ matrix A is called nilpotent if $A^m=0$ for some positive integer in. Examples are triangularmatrices whose entries on the diagonal are all 0. Consider a nilpolent $n\times n$ matrix A. and choose the smallest number in such that $A^m=0$ . Pick a vector $\stackrel{\to }{v}$ in ${ℝ}^n$ such that $A^{m-1}\stackrel{\to }{v}\ne \stackrel{\to }{0}$ . Show that the vectors $\stackrel{\to }{v},A\stackrel{\to }{v},A^2\stackrel{\to }{v},\mathrm{...},A^{m-1}\stackrel{\to }{v}$ are linearly independent. Hint: Consider a relation $c_0\stackrel{\to }{v}+c_{1}A\stackrel{\to }{v}+c_2{A}^2\stackrel{\to }{v}+\cdots +c_{m-1}{A}^{m-1}\stackrel{\to }{v}=\stackrel{\to }{0}$ . Multiply both sides of the equationwith $A^{m-1}$ to show that $c_0=0$ . Next, show that $c_1=0$ ,and so on.