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Section 2.4 Planar Algebra and Geometry

Exercises Exercises

1. Matrices and complex functions.

Let \(R_{\theta}\) denote the rotation of the plane about the origin by \(\theta\) revolutions. For example, \(R_{1/2}\) is the linear map \(\R^2\to \R^2\) given by the matrix \(\twotwo{-1}{0}{0}{-1}\text{.}\) Alternatively, \(\R_{1/2}\) is the map \(\C\to\C\) given by \(R_{1/2}(z)=-z\text{.}\) 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_{1/4},R_{1/2},R_{3/4},F_H,F_V,F_D,F_{D'}\}. \end{equation*}

2. Roots of unity.

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

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

4. 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 \(\mu_\alpha,t_{\beta},r\colon \extR\to\extR\text{,}\) 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*}
Any of these functions, and any possible composition of these types of functions (for all possible values of \(\alpha,\beta\) with \(\alpha\neq 0\)), is called a real linear fractional transformation of the extended real numbers.
  1. Show that \(\mu_{\alpha},\tau_{\beta},\rho\) are invertible functions.
  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 \(T(x)=\frac{ax+b}{cx+d}\text{.}\) 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 \(T(x)=\frac{ax+b}{cx+d}\text{.}\) 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 \(T(x)=\frac{ax+b}{cx+d}\) is called a linear fractional transformation on the extended reals. Given a linear fractional transformation \(T(x)=\frac{ax+b}{cx+d}\text{,}\) let \(M(T)=\twotwo{a}{b}{c}{d}\text{.}\) We will call \(M(T)\) the matrix of coefficients of \(T\text{.}\) Suppose that \(S\) is another linear fractional transformation with \(M(S)=\twotwo{e}{f}{g}{h}\text{.}\) Show that \(M(T\of S)=M(T)M(S)\) and \(M(T^{-1})=(M(T))^{-1}\) (the previous two parts of this problem show that \(T\) is invertible).
  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 if \(M\mathbf{v}=\lambda \mathbf{v}\text{.}\) Show that if \(x_0\) is a fixed point for \(T(x)=\frac{ax+b}{cx+d}\text{,}\) then \(\begin{bmatrix}x_0\\1\end{bmatrix}\) is an eigenvector for \(M(T)=\twotwo{a}{b}{c}{d}\) with eigenvalue \(cx_0+d\text{.}\)