- 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
2.4 Exercises
In each of Exercises 11 through 15, a transformation $T: V_2 \rightarrow V_2$ is described as indicated. In each case determine whether $T$ is linear. If $T$ is linear, describe its null space and range, and compute its nullity and rank.
13. $\quad$ $T$ maps every point onto the point $(1, 1).$
Solution. $\quad$ Recall from Section 2.1 that if $V$ and $W$ are linear spaces, then a function $T: V \to W$ is called a linear transformation from $V$ into $W$ if it has the following two properties:
$\quad$ (a) $\quad T(x + y) = T(x) + T(y) \quad$ for all $x$ and $y$ in $V,$
$\quad$ (b) $\quad T(cx) = cT(x) \quad$ for all $x$ in $V$ and all scalars $c.$
Let $X$ and $Y$ be any two elements of $V_2.$ Then, $T(X + Y) = (1, 1)$ and $T(X) + T(Y) = (2, 2).$ Thus, $T(X + Y) \neq T(X) + T(Y)$ and $T$ is nonlinear. $\quad \blacksquare$