2.E: Exercises for Chapter 2
- Page ID
- 269
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Calculational Exercises
1. Express the following complex numbers in the form \(x + yi\) for \(x, y \in \mathbb{R}:\)
(a) \((2 + 3i) + (4 + i)\)
(b) \((2 + 3i)^2 (4 + i)\)
(c) \(\frac{2+3i}{4+i}\)
(d) \(\frac{1}{i}+\frac{3}{1+i}\)
(e) \((−i)^{−1}\)
(f) \((−1 + i \sqrt{3})^3\)
2. Compute the real and imaginary parts of the following expressions, where \(z\) is the
complex number \(x + yi\) and \(x, y \in \mathbb{R}:\)
(a) \(\frac{1}{z^2}\)
(b) \(\frac{1}{3z+2}\)
(c) \(\frac{z+1}{2z-5}\)
(d) \(z^3\)
3. Find \(r > 0\) and \(\theta \in [0, 2\pi) \) such that \((1 − i)/ 2 = re^{i \theta}.\)
4. Solve the following equations for \(z\) a complex number:
(a) \(z^5 − 2 = 0\)
(b) \(z^4 + i = 0\)
(c) \(z^6 + 8 = 0\)
(d) \(z^3 − 4i = 0\)
5. Calculate the
(a) complex conjugate of the fraction \((3 + 8i)^4 /(1 + i)^10 .\)
(b) complex conjugate of the fraction \((8 − 2i)^10 /(4 + 6i)^5 .\)
(c) complex modulus of the fraction \(i(2 + 3i)(5 − 2i)/(−2 − i).\)
(d) complex modulus of the fraction \((2 − 3i)^2 /(8 + 6i)^2 .\)
6. Compute the real and imaginary parts:
(a) \(e^{2+i}\)
(b) \(sin(1 + i)\)
(c) \(e^{3−i}\)
(d) \(cos(2 + 3i)\)
7. Compute the real and imaginary part of \(e^{e^{z}}\) for \(z \in \mathbb{C}.\)
Proof-Writing Exercises
1. Let \(a \in \mathbb{R}\) and \(z, w \in \mathbb{C}.\) Prove that
(a) \( Re(az) = aRe(z)\) and \( Im(az) = aIm(z).\)
(b) \( Re(z + w) = Re(z) + Re(w)\) and \( Im(z + w) = Im(z) + Im(w).\)
2. Let \(z \in \mathbb{C}.\) Prove that \( Im(z) = 0\) if and only if \( Re(z) = z.\)
3. Let \(z, w \in \mathbb{C}.\) Prove the parallelogram law \(|z − w|^2 + |z + w|^2 = 2(|z|^2 + |w|^2).\)
4. Let \(z, w \in \mathbb{C}\) with \(\bar{z}w \neq 1\) such that either \(|z| = 1\) or \(|w| = 1.\) Prove that \( \left| \frac{z−w}{1 − \bar{z}w} \right| =1. \)
5. For an angle \(\theta \in [0, 2\pi),\) find the linear map \(f_\theta : \mathbb{R}^2 \rightarrow \mathbb{R}^2\), which describes the rotation by the angle \(\theta\) in the counterclockwise direction.
Hint: For a given angle \(\theta\), find \(a, b, c, d \in \mathbb{R}\) such that \(f_\theta (x_1 , x_2 ) = (ax_1 +bx_2 , cx_1 +dx_2 ).\)
Contributors
- Isaiah Lankham, Mathematics Department at UC Davis
- Bruno Nachtergaele, Mathematics Department at UC Davis
- Anne Schilling, Mathematics Department at UC Davis
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