- 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 3 through 12, a function $T: V_2 \rightarrow V_2$ is defined by the formula given for $T(x, y),$ where $(x, y)$ is an arbitrary point in $V_2.$ In each case, determine whether $T$ is one-to-one on $V_2.$ If it is, describe its range $T(V_2);$ for each point $(u, v)$ in $T(V_2),$ let $(x, y) = T^{-1}(u, v)$ and give formulas for determining $x$ and $y$ in terms of $u$ and $v.$
6.$\quad$ $T(x, y) = (x, x).$
Solution. $\quad$ $T$ is not a one-to-one transformation on $V_2.$ For example, let $A = (1, 1)$ and $B = (1, -1)$ be two points in $V_2.$ We have $A \neq B,$ but $T(A) = T(B). \quad \blacksquare$