# Not Just Calculus: First-Year Introduction to Upper-Level Mathematics

## Section2.4Problems

### Subsection2.4.1Some basic algebra and geometry in the plane

#### ExercisesExercises

##### 1.Matrices and complex functions.
Let $$R_{\theta}$$ denote the rotation of the plane about the origin by $$\theta$$ radians. For example, $$R_{\pi}$$ is the linear map $$\R^2\to \R^2$$ given by the matrix $$\twotwo{-1}{0}{0}{-1}\text{.}$$ Alternatively, $$\R_{\pi}$$ is the map $$\C\to\C$$ given by $$R_{\pi}(z)=-z\text{.}$$ Let $$F_H,F_V,F_D,F_{D'}$$ be the reflections across the lines $$H: y=0\text{,}$$ $$V: x=0\text{,}$$ $$D: y=-x\text{,}$$ and $$D': y=x\text{.}$$ For example $$F_H$$ is the linear map $$\R^2\to \R^2$$ given by the matrix $$\twotwo{1}{0}{0}{-1}\text{.}$$ Alternatively, $$F_H$$ is the map $$\C\to\C$$ given by $$F_H(z)=z^{\ast}\text{.}$$ Find linear maps and complex functions for all eight of the mappings in the set
\begin{equation*} D_4=\{R_0,R_{\pi/2},R_{\pi},R_{3\pi/2},F_H,F_V,F_D,F_{D'}\}. \end{equation*}
For each complex function $$f\text{,}$$ find a constant $$\alpha$$ so that $$f(z)$$ may be written in the form $$f(z)=\alpha z$$ or in the form $$f(z)=\alpha z^\ast\text{.}$$
Hint.
For the complex functions for $$F_D$$ and $$F_{D'}\text{,}$$ use part c of Exercise 2.2.5.
##### 2.Orthogonal projections onto lines.
Let $$\mathbf{u}$$ be a vector in $$\R^2$$ such that $$\left\Vert\mathbf{u}\right\Vert=1\text{,}$$ let $$\ell$$ be the line
\begin{equation*} \ell=\{\alpha \mathbf{u}\colon \alpha \in \R\} \end{equation*}
and let $$\mathbf{v}=\overrightarrow{OQ}$$ be another vector in $$\R^2\text{,}$$ where $$O=(0,0)$$ is the origin and $$Q$$ is some point in the plane. Let $$P$$ be the point on $$\ell$$ such that $$\triangle OPQ$$ is a right triangle. Use (2.1.10) and draw a sketch to explain why
\begin{equation*} \overrightarrow{OP}=(\mathbf{u}\cdot \mathbf{v})\mathbf{u}. \end{equation*}
The vector $$\overrightarrow{OP}$$ is called the (orthogonal) projection of $$\mathbf{v}$$ onto $$\mathbf{u}$$, denoted $$\proj_{\mathbf{u}}\mathbf{v}$$.

### Subsection2.4.2Roots of unity

Let $$n$$ be a positive integer. A (complex) $$n$$-th root of unity is a complex number $$z$$ such that $$z^n=1\text{.}$$

#### ExercisesExercises

##### 1.
Show that if $$z$$ is an $$n$$-th root of unity, then $$|z|=1\text{.}$$
##### 2.
Find all of the $$n$$-th roots of unity for $$n=3,6\text{.}$$ Sketch their locations on the unit circle.
##### 3.
Let $$\omega=E(2\pi/n)\text{.}$$ Show that if $$z$$ is an $$n$$-th root of unity, then $$z=\omega^k$$ for some integer $$k\text{.}$$
##### 4.
Let $$k$$ be an integer. Show that $$\omega^k=\omega^r$$ for some $$r$$ in the range $$0\leq r\leq n-1\text{.}$$
##### 5.
An $$n$$-th root of unity $$z$$ is called primitive if all $$n$$-th roots of unity can be written as powers of $$z\text{.}$$ Find all of the primitive roots of unity for $$n=3,6,10\text{.}$$
##### 6.
Let $$C_n$$ denote the set of $$n$$-th roots of unity, and let $$\mu\colon \Z_n\to C_n$$ be given by $$\mu(k)=\omega^k\text{.}$$ Show that $$\mu$$ is one-to-one and onto, and that $$\mu(k)\mu(\ell)=\mu(k+_n\ell)$$ for all $$k,\ell$$ in $$\Z_n\text{.}$$

### Subsection2.4.3Linear fractional transformations of the extended real numbers

We call the set
\begin{equation*} \extR=\R\cup \{\infty\} \end{equation*}
the extended real numbers, where $$\infty$$ is an element that is not a real number. Let $$\alpha\neq 0$$ be a real number and let $$\beta$$ be any real number. We define functions $$\mu_\alpha,\tau_{\beta},\rho\colon \extR\to\extR\text{,}$$ as follows.
\begin{align*} \mu_{\alpha}(x)\amp = \begin{cases} \alpha x \amp x\in \R\\ \infty \amp x=\infty \end{cases}\\ \tau_{\beta}(x) \amp = \begin{cases} x+\beta \amp x\in \R\\ \infty \amp x=\infty \end{cases}\\ \rho(x) \amp = \begin{cases} 1/x \amp x\in \R,x\neq 0\\ \infty \amp x=0\\ 0 \amp x=\infty \end{cases} \end{align*}

#### ExercisesExercises

##### 1.
Show that $$\mu_{\alpha},\tau_{\beta},\rho$$ are invertible functions. Say where you use the assumption that $$\alpha\neq 0\text{.}$$
##### 2.
Let $$a,b,c,d$$ be real numbers with $$c\neq 0$$ and $$ad-bc\neq 0\text{.}$$ Show that the composition
\begin{equation*} \mu_{(1/c)}\of\tau_a\of \mu_{(bc-ad)}\of\rho\of \tau_d\of \mu_c \end{equation*}
is given by
$$T(x)=\begin{cases} \frac{ax+b}{cx+d} \amp \text{ if }x\in \R,x\neq -d/c\\ \infty \amp \text{ if } x= -d/c\\ a/c \amp \text{ if }x=\infty. \end{cases}\tag{2.4.1}$$
Conclude that $$T\colon \extR\to\extR$$ is invertible.
##### 3.
Let $$a,b,c,d$$ be real numbers with $$c=0$$ and $$ad-bc\neq 0\text{.}$$ Show that $$d\neq 0$$ and that the composition $$\tau_{(b/d)}\of \mu_{a/d}$$ is given by
$$T(x)=\begin{cases} \frac{ax+b}{cx+d} \amp \text{ if } x\neq \infty\\ \infty \amp \text{ if } x= \infty. \end{cases}\tag{2.4.2}$$
Conclude that $$T\colon \extR\to\extR$$ is invertible.
##### 4.
Let $$a,b,c,d$$ be real numbers with $$ad-bc\neq 0\text{.}$$ The function $$T\colon \extR\to \extR$$ given by $$T(x)=\frac{ax+b}{cx+d}\text{,}$$ with exceptional values specified by (2.4.1) and (2.4.2) for the cases $$c\neq 0\text{,}$$ $$c=0\text{,}$$ respectively, is called a linear fractional transformation on the extended reals. Given an invertible matrix $$\twotwo{a}{b}{c}{d}\text{,}$$ let $$T_M$$ denote the linear fractional transformation given by
\begin{equation*} T_M(x) = \frac{ax+b}{cx+d}. \end{equation*}
Suppose that $$N=\twotwo{e}{f}{g}{h}$$ is another invertible matrix, so that $$T_N(x)=\frac{ex+f}{fx+h}\text{.}$$ Let $$\Id$$ denote the identity matrix, and let $${\rm id}$$ denote the identity function $${\rm id}\colon \extR\to \extR\text{.}$$ Show the following.
1. $$\displaystyle T_{MN}=T_M\of T_N$$
2. $$\displaystyle T_{\Id}={\rm id}$$
3. $$\displaystyle (T_M)^{-1}=T_{(M^{-1})}$$
##### 5.
Let $$s\colon S^1\to \extR$$ be the stereographic projection function, given by
\begin{equation*} s(x,y)=\begin{cases}\frac{x}{1-y}\amp y\neq 1\\ \infty \amp y=1. \end{cases} \end{equation*}
Show that $$s$$ is a bijection.
##### 6.
Given a function $$f\colon \extR\to\extR\text{,}$$ the lift of $$f$$ by $$s$$ is the function $$s^{-1}\of f\of s\colon S^1\to S^1\text{.}$$ Show that the lift of the linear fractional transformation $$T(x)=1/x$$ is given by $$(x,y)\to (x,-y)$$ and that the lift of $$S(x)=-x$$ is given by $$(x,y)\to (-x,y)\text{.}$$
##### 7.
An element $$x_0\in A$$ is called a fixed point of a function $$f\colon A\to A$$ if $$f(x_0)=x_0\text{.}$$ Find all the fixed points of the linear fractional transformation $$T(x)=\frac{6x+2}{3x+3}\text{,}$$ and find the fixed points of the lift $$s^{-1}\of T\of s\text{.}$$
##### 8.
Let $$M\colon \R^2\to \R^2$$ be a linear map. A nonzero vector $$\mathbf{v}\in \R^2$$ is called an eigenvector for $$M$$ with eigenvalue $$\lambda\in \R$$ if $$M\mathbf{v}=\lambda \mathbf{v}\text{.}$$ Show that if $$x_0$$ is a fixed point for a linear fractional transformation $$T(x)=\frac{ax+b}{cx+d}\text{,}$$ then $$\begin{bmatrix}x_0\\1\end{bmatrix}$$ is an eigenvector for $$M(T)=\twotwo{a}{b}{c}{d}$$ with eigenvalue $$cx_0+d\text{.}$$

### Subsection2.4.4Integral points on the unit circle

We define the set $$\mathcal{I}\text{,}$$ called the set of integral points in the unit circle $$S^1\text{,}$$ by
\begin{equation*} \mathcal{I} = \{E(k)\colon k\in \Z\}. \end{equation*}
This subsection is an exploration of how integral points are distributed in the unit circle. We begin with a Lemma.

#### Checkpoint2.4.2.

Let $$a,b$$ be real numbers with $$a\lt b\text{.}$$ We define the open interval $$U_{a,b}$$ of the unit circle to be the set
\begin{equation*} U_{a,b}=\{E(t)\colon a\lt t\lt b\}. \end{equation*}
We say that a subset $$X\subseteq S^1$$ is dense in $$S^1$$ if every open interval of $$S^1$$ contains an element of $$X\text{.}$$

#### Proof.

(This is an outline of the main points of the proof. You will complete the details in the exercises below.) Let $$a,b$$ be given, with $$a\lt b\text{,}$$ and let $$\epsilon=b-a\text{.}$$ If $$\epsilon \geq 2\pi\text{,}$$ then $$U_{a,b}=S^1\text{,}$$ so $$U_{a,b}$$ contains an element of $$\mathcal{I}$$ because $$U_{a,b}$$ contains all of $$\mathcal{I}\text{.}$$ For the remainder of the proof, we assume $$\epsilon\lt 2\pi\text{.}$$
1. Choose a finite collection of open intervals $$\{U_k\}_{k=1}^N\text{,}$$ where $$N$$ is a positive integer and each $$U_k$$ is an arc of length $$\epsilon\text{,}$$ that is,
\begin{equation*} U_k =U_{a_k,a_k+\epsilon} \end{equation*}
for some real number $$a_k\text{,}$$ and such that
\begin{equation*} S^1=\bigcup_{k=1}^N U_k. \end{equation*}
2. At least one of the intervals $$U_k$$ in the previous part must contain at least two elements of $$\mathcal{I}\text{,}$$ say $$E(m)$$ and $$E(n)\text{,}$$ with $$E(m)\neq E(m)\text{.}$$
3. For the two points $$E(m),E(n)$$ in the previous part, choose real numbers $$s,t$$ such that $$0\leq s\lt t\lt 2\pi\text{,}$$ such that $$\{E(m),E(n)\}=\{E(s),E(t)\}\text{.}$$ Without loss of generality, we may assume $$E(m)=E(s)$$ and $$E(n)=E(t)\text{.}$$
4. Let $$t_0=t-s\text{,}$$ so that $$0\lt t_0\lt \epsilon\text{.}$$ We have
\begin{equation*} E(n)E(-m)=E(n-m)=E(t-s)=E(t_0). \end{equation*}
5. Choose an integer $$k$$ so that $$a\lt kt_0 \lt b\text{.}$$ Now we have $$E(kt_0)=E(k(n-m))\in U_{a,b}\cap \mathcal{I}\text{,}$$ as desired.

#### ExercisesExercises

##### 1.
Give specific details for how one can choose the intervals $$U_k$$ in part 1 of the proof of Proposition 2.4.3. How do you know what integer $$N$$ is needed? Write explicit expressions for the numbers $$a_k\text{.}$$
##### 2.
Justify the claim made in part 2 of the proof. Why must there be an interval $$U_k$$ that contains two elements of $$\mathcal{I}\text{.}$$ (Hint: what if this were not the case?)
##### 3.
Draw sketches to illustrate what is going on in part 3 of the proof. Why is the phrase "without loss of generality" needed?
##### 4.
In part 4 of the proof, how do we know that $$0\lt t_0\lt \epsilon\text{?}$$ Justify each of the "equals" signs in the displayed equation.
##### 5.
In part 5 of the proof, how do we know that it is possible to choose the integer $$k\text{?}$$ (Hint: what if it were not possible?) Justify the "equals" sign and the "is an element of" sign in the last sentence.