Chapter 3.4, Problem 81E

Consider the linear transformation $T\left[\begin{array}{l}{x}_1\\ x_2\\ x_3\end{array}\right]=\left[\begin{array}{l}{x}_2\\ x_3\\ x_2+x_3\end{array}\right]$ from ${ℝ}^3$ to ${ℝ}^3$ .
a. Find all vectors of the form $\stackrel{\to }{x}=\left[\begin{array}{l}1\\ x_2\\ x_3\end{array}\right]$ such that $T\left(\stackrel{\to }{x}\right)$ is a scalar multiple of $\stackrel{\to }{x}$ . Be prepared to dealwith irrational numbers.

b. Find a basis of ${ℝ}^3$ such that the matrix B of Tis diagonal.