Problems

Section 2.4 Problems

Subsection 2.4.1 Some basic algebra and geometry in the plane

Exercises Exercises

1. Matrices and complex multiplication.

Section 2.3 begins with a one-to-one correspondence \(\C\leftrightarrow \R^2\) given by \(z\leftrightarrow (\re(z),\im(z))\text{.}\) The mapping that sends \(z\in \C\) to \((\re(z),\im(z))\in \R^2\) is sometimes called "realify". Now let \(z\) be a complex number, let \(\mu_z \colon \C\to \C\) be the map given by \(\mu_z(w)=zw\text{,}\) and let \(L_z\colon \R^2\to \R^2\) be given by

\begin{equation*} L_z = \realify \circ \; \mu_z \circ \realify^{-1}. \end{equation*}

Show that \(L_z\) is a linear map, and find its matrix.

Hint.

Use Proposition 2.2.4.

2. 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.

3. 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 \(\proj_{\mathbf{u}}\mathbf{v}\).

Subsection 2.4.2 Roots of unity

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

Exercises Exercises

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 \(\Z_n\text{.}\)

Subsection 2.4.3 Linear 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*}

Exercises Exercises

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

\begin{equation} 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} \end{equation}

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

\begin{equation} 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} \end{equation}

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.

\(\displaystyle T_{MN}=T_M\of T_N\)
\(\displaystyle T_{\Id}={\rm id}\)
\(\displaystyle (T_M)^{-1}=T_{(M^{-1})}\)

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 \(cx_0+d\text{.}\)

Subsection 2.4.4 Integral 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.

Lemma 2.4.1.

If \(E(m)=E(n)\) for integers \(m,n\text{,}\) then \(m=n\text{.}\)

Checkpoint 2.4.2.

Prove Lemma 2.4.1. Hint: examine your solution to Exercise 1.8.2.3.

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 \(X\text{.}\)

Proposition 2.4.3.

The set \(\mathcal{I}\) is dense in \(S^1\text{.}\)

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 \(\epsilon\lt 2\pi\text{.}\)

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*}
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 \(E(m)\neq E(m)\text{.}\)
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 \(E(n)=E(t)\text{.}\)
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*}
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.

Exercises Exercises

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.

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