3.E: Exercises for Chapter 3
- Page ID
- 303
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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. Let \(n \in \mathbb{Z}_+\) be a positive integer, let \(w_0 , w_1 ,\ldots, w_n \in \mathbb{C}\) be distinct complex numbers, and let \(z_0 , z_1 ,\ldots, z_n \in \mathbb{C}\) be any complex numbers. Then one can prove that there is a unique polynomial \(p(z)\) of degree at most \(n\) such that, for each \(k \in \{0, 1, . . . , n\}, p(w_k ) = z_k.\)
(a) Find the unique polynomial of degree at most \(2\) that satisfies \(p(0) = 0, p(1) = 1,\) and \(p(2) = 2.\)
(b) Can your result in Part (a) be easily generalized to find the unique polynomial of degree at most \(n\) satisfying \(p(0) = 0, p(1) = 1, \ldots , p(n) = n\)?
2. Given any complex number \(\alpha \in \mathbb{C},\) show that the coefficients of the polynomial
\[(z − \alpha)(z − \bar{\alpha})\]
are real numbers.
Proof-Writing Exercises
1. Let \(m, n \in \mathbb{Z}_+\) be positive integers with \(m \leq n\). Prove that there is a degree n polynomial \(p(z)\) with complex coefficients such that \(p(z)\) has exactly m distinct roots.
2. Given a polynomial \(p(z) = a_n z^n + \cdots + a_1 z + a_0\) with complex coefficients, define the conjugate of \(p(z)\) to be the new polynomial
\[ \bar{p}(z) = \bar{a_n} z^n + \cdots + \bar{a_1}z + a_0. \]
(a) Prove that \(\bar{p(z)} = \bar{p}(\bar{z}).\)
(b) Prove that \(p(z)\) has real coefficients if and only if \(\bar{p}(z) = p(z).\)
(c) Given polynomials \(p(z), q(z),\) and \(r(z)\) such that \(p(z) = q(z)r(z),\) prove that \(\bar{p}(z) = \bar{q}(z)\bar{r}(z).\)
3. Let \(p(z)\) be a polynomial with real coefficients, and let \( \alpha \in \mathbb{C}\) be a complex number.
Prove that \(p(\alpha) = 0\) if and only \(p(\bar{\alpha}) = 0.\)
Contributors
- Isaiah Lankham, Mathematics Department at UC Davis
- Bruno Nachtergaele, Mathematics Department at UC Davis
- Anne Schilling, Mathematics Department at UC Davis
Both hardbound and softbound versions of this textbook are available online at WorldScientific.com.