Chapter 3.7, Problem 29E

In Exercise 25-30, a linear transformation $T$ is given. In each case find a matrix $A$ such that $T\left(\text{x}\right)=A\left(\text{x}\right)$. Also describe the null space and the range of $T$ and give the rank and nullity of $T$.

$T:R^3\to R^2$ defined by

$T\left(\left[\begin{array}{c}{x}_1\\ x_2\\ x_3\end{array}\right]\right)=\left[\begin{array}{c}{x}_1-x_2\\ x_2-x_3\end{array}\right]$