- 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.
15. $\quad$ $T$ maps each point with polar coordinates $(r, \theta)$ onto the point with polar coordinates $(r, 2\theta)$. Also, $T$ maps $O$ onto itself.
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 $c.$
The transformation is nonlinear. We will demonstrate this with a counterexample.
$\quad$ Counterexample. $\quad$ Let $X = (1, 0)$ and $Y = (-1, 0)$ be the points in $V_2.$ This corresponds to $X$ and $Y$ having polar coordinates $(1, 0)$ and $(1, \pi)$ respectively. Then, applying $T$ to $X$ and $Y,$ we get polar coordiates $(1, 0)$ and $(1, 2\pi),$ giving us \begin{align*} T(X) &= T(Y) = (1, 0) \end{align*} Taking the sum of $X$ and $Y,$ we find that $X + Y = (1, 0) + (-1, 0) = O.$ And since $T$ maps $O$ to itself, we have: \begin{align*} T(X + Y) &= T(O) \\ &= O \\ &\neq T(X) + T(Y) \quad \blacksquare \end{align*}