- 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 $w.$
20.$\quad$ $T(x, y, z) = (x + y, y + z, x + z).$
Solution. $\quad$ Let $A = (x, y, z)$ and $B = (p, q, r)$ be two points in $V_3.$ Then \begin{align*} T(A) &= (x + y, y + z, x + z) \\ T(B) &= (p + q, q + r, p + r) \end{align*} If $T(A) = T(B)$ then we have $x + y = p + q$ and $x + z = p + r.$ Subtracting the second equation from the first, we find that $y - z = q - r.$ Adding the equation $y + z = p + q,$ we get $2y = 2q,$ or $y = q.$ From this, we find that $x = p$ and $z = r.$ As such, $T(A) = T(B),$ implies $A = B,$ which means $T$ one-to-one on $V_3.$ The range of $T$ is $V_3,$ and for each point $(u, v, w)$ in $T(V_3),$ if we let $(x, y, z) = T^{-1}(u, v, w),$ we have the formulas \begin{align*} x &= \frac{1}{2}\left(u - v + w \right), \quad y = \frac{1}{2}\left(v - w + u \right),\quad z = \frac{1}{2}\left(w - u + v \right). \quad \blacksquare \end{align*}