- Calculus, Volume 2: Multi-Variable Calculus and Linear Algebra with Applications to Differential Equations and Probability
- Tom M. Apostol
- Second Edition
- 1991
- 978-1-119-49676-2
2.8 Exercises
In each of Exercises 13 through 20, a function $T: V_3 \rightarrow V_3$ is defined by the formula given for $T(x, y, z),$ where $(x, y, z)$ is an arbitrary point in $V_3.$ In each case, determine whether $T$ is one-to-one on $V_3.$ If it is, describe its range $T(V_3);$ for each point $(u, v, w)$ in $T(V_3),$ let $(x, y, z) = T^{-1}(u, v, w)$ and give formulas for determining $x,$ $y,$ and $z$ in terms of $u,$ $v,$ and $w.$
14.$\quad$ $T(x, y, z) = (x, y, 0).$
Solution. $\quad$ $T$ is not one-to-one on $V_3.$ For example, let $A = (1, 1, 1)$ and $B = (1, 1, 0)$ be two points in $V_3.$ Even though $A \neq B,$ we have $T(A) = T(B). \quad \blacksquare$