Mathematical Immaturity

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.

11. $\quad$ $T$ rotates every point through the same angle $\phi$ about the origin. That is, $T$ maps a point with polar coordinates $(r, \theta)$ onto the point with polar coordinates $(r, \theta + \phi)$, where $\phi$ is fixed. 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 $c.$

Let $(r_x, \theta_x)$ and $(r_y, \theta_y)$ be the polar coordinates of points $x$ and $y,$ respectively, and let $c$ be any scalar. Because $x$ and $y$ are elements of $V_2,$ we can express them in terms of their plane cooridnates as follows: \begin{align*} x &= |r_x|(\cos \theta_x, \sin \theta_x), \quad y = |r_y|(\cos \theta_y, \sin \theta_y) \end{align*} Then, if $T(x) = (r_x, \theta_x + \phi)$ and $T(y) = (r_y, \theta_y + \phi),$ we have: \begin{align*} T(x) &= |r_x|\left[\cos (\theta_x + \phi), \sin (\theta_x + \phi)\right] \\ T(y) &= |r_y|\left[\cos (\theta_y + \phi), \sin (\theta_y + \phi)\right] \end{align*} If we define addition and scalar multiplication component-wise, we have: \begin{align*} T(x + y) &= T(|r_x|\cos \theta_x + |r_y|\cos \theta_y, |r_x|\sin \theta_x + |r_y|\sin \theta_y) \\ &= \left[|r_x|\cos (\theta_x + \phi) + |r_y|\cos (\theta_y + \phi), |r_x|\sin (\theta_x + \phi) + |r_y|\sin (\theta_y + \phi)\right] \\ &= [|r_x|\cos (\theta_x + \phi), |r_x|\sin(\theta_x + \phi)] + [|r_y|\cos (\theta_y + \phi), |r_y|\sin(\theta_y + \phi)] \\ &= T(x) + T(y) \\ \\ T(cx) &= T(c|r_x|\cos \theta_x, c|r_x|\sin \theta_x) \\ &= [c|r_x|\cos (\theta_x + \phi), c|r_x|\sin (\theta_x + \phi)] \\ &= c[|r_x|\cos (\theta_x + \phi), |r_x|\sin (\theta_x + \phi)] \\ &= cT(x) \end{align*} Thus, for $x$ in $V_2$ with polar coordinates $(r, \theta)$, the transformation $T(x) = (r, \theta + \phi)$ is linear. It has null space $N(T) = \{(0, 0)\}$ and range $V_2,$ making its nullity $0$ and its rank $2. \quad \blacksquare.$