- 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 $w.$
13.$\quad$ $T(x, y, z) = (z, y, x).$
Solution. $\quad$ Let $A = (x, y, z)$ and $B = (p, q, r)$ be points in $V_3$ with transformations $T(A) = (z, y, x)$ and $T(B) = (r, q, p).$ Then, if $T(A) = T(B),$ we have \begin{align*} z = r, \quad y = q, \quad x = p \end{align*} In other words, if $T(A) = T(B),$ then $A = B,$ making $T$ one-to-one on $V_3.$ The range of $T$ is $V_3,$ and for each point $(u, v, w)$ in $T(V_3),$ we can determine the point $(x, y, z) = T^{-1}(u, v, w)$ by setting \begin{align*} x = w, \quad y = v, \quad z = u \quad \blacksquare \end{align*}