Chapter 3.4, Problem 58E

Consider a $3\times 3$ matrix A and a vector $\stackrel{\to }{v}$ in ${ℝ}^3$ suchthat $A^3\stackrel{\to }{v}=\stackrel{\to }{0}$ , but $A^2\stackrel{\to }{v}\ne \stackrel{\to }{0}$ .
a. Show that the vectors $A^2\stackrel{\to }{v},A\stackrel{\to }{v},\stackrel{\to }{v}$ form a basis of ${ℝ}^3$ . Him’: It suffices lo show linear independence.Consider a relation $c_1{A}^2\stackrel{\to }{v}+c_{2}A\stackrel{\to }{v}+c_3\stackrel{\to }{v}=\stackrel{\to }{0}$ andmultiply by $A^2$ to show that $c_3=0$ .
b. Find the matrix of the transformation $T\left(\stackrel{\to }{x}\right)=A\stackrel{\to }{x}$ with respect to the basis $A^2\stackrel{\to }{v},A\stackrel{\to }{v},\stackrel{\to }{v}$ .