<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\documentclass{article}

% This is calling style.sty
%%%%%%%%%%%%%%%%%% Do not remove the following %%%%%%%%%%%%%
\usepackage{fullpage}
\usepackage{algpseudocode}
\usepackage{color}
\usepackage{float,graphicx,tabularx}
%\usepackage[labelformat=simple]{subcaption}
\usepackage{eqparbox}
\usepackage{amsmath,amsthm,amsfonts,amssymb}
\usepackage{bbm}
\usepackage{leftidx}
\usepackage{mathtools}
\usepackage{environ}
%\usepackage{semantic} % conflict with lstlisting
\usepackage[version=4]{mhchem} % chemical formula
\usepackage{hyperref}
\usepackage{parskip}
\hypersetup{
     colorlinks   = true,
     citecolor    = blue
}
%\renewcommand\thesubfigure{(\alph{subfigure})}

\definecolor{darkgreen}{RGB}{0,128,0}

\DeclarePairedDelimiter\abkt{\langle}{\rangle}
\DeclarePairedDelimiter\cbkt{\lbrace}{\rbrace}
\DeclarePairedDelimiter\rbkt{\lparen}{\rparen}
\DeclarePairedDelimiter\sbkt{\lbrack}{\rbrack}
\DeclarePairedDelimiter\abs{\lvert}{\rvert}
\DeclarePairedDelimiter\norm{\lVert}{\rVert}
\DeclarePairedDelimiter\floor{\lfloor}{\rfloor}
\DeclarePairedDelimiter\ceiling{\lceil}{\rceil}
\DeclarePairedDelimiterX\set[2]{\lbrack}{\rbrack}{#1 \colon #2}

\newcommand{\clapover}[2]{\overbrace{#1}_{\mathclap{#2}}}
\newcommand{\clapunder}[2]{\underbrace{#1}_{\mathclap{#2}}}

\DeclareMathOperator*{\argmin}{arg\,min}
\DeclareMathOperator*{\argmax}{arg\,max}
\DeclareMathOperator{\supp}{supp}
\newcommand{\defeq}{\stackrel{\text{def}}=}

\DeclareMathOperator{\matrixtranspose}{T}
\newcommand{\T}{{\matrixtranspose}}
\DeclareMathOperator{\frobeniusnorm}{F}
\newcommand{\Fro}{{\frobeniusnorm}}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\sgn}{sgn}

\DeclareMathOperator*{\E}{E}
\DeclareMathOperator*{\Var}{Var}
\DeclareMathOperator*{\Cov}{Cov}
\DeclareMathOperator*{\Corr}{Corr}
\DeclareMathOperator{\BinDist}{Bin}
\DeclareMathOperator{\GaussDist}{N}
\DeclareMathOperator{\PoisDist}{Pois}
\DeclareMathOperator{\UnifDist}{U}
\DeclareMathOperator{\erf}{erf} % error function.

\DeclareMathOperator{\df}{d} % Differential form.
\DeclareMathOperator{\round}{\partial}

\DeclareMathOperator{\area}{area}
\DeclareMathOperator{\vol}{vol}

\newcommand{\eps}{\varepsilon}
\newcommand{\haat}{\widehat}

\newcommand{\bbA}{{\mathbb{A}}}
\newcommand{\bbB}{{\mathbb{B}}}
\newcommand{\bbC}{{\mathbb{C}}}
\newcommand{\bbD}{{\mathbb{D}}}
\newcommand{\bbE}{{\mathbb{E}}}
\newcommand{\bbF}{{\mathbb{F}}}
\newcommand{\bbG}{{\mathbb{G}}}
\newcommand{\bbH}{{\mathbb{H}}}
\newcommand{\bbI}{{\mathbb{I}}}
\newcommand{\bbJ}{{\mathbb{J}}}
\newcommand{\bbK}{{\mathbb{K}}}
\newcommand{\bbL}{{\mathbb{L}}}
\newcommand{\bbM}{{\mathbb{M}}}
\newcommand{\bbN}{{\mathbb{N}}}
\newcommand{\bbO}{{\mathbb{O}}}
\newcommand{\bbP}{{\mathbb{P}}}
\newcommand{\bbQ}{{\mathbb{Q}}}
\newcommand{\bbR}{{\mathbb{R}}}
\newcommand{\bbS}{{\mathbb{S}}}
\newcommand{\bbT}{{\mathbb{T}}}
\newcommand{\bbU}{{\mathbb{U}}}
\newcommand{\bbV}{{\mathbb{V}}}
\newcommand{\bbW}{{\mathbb{W}}}
\newcommand{\bbX}{{\mathbb{X}}}
\newcommand{\bbY}{{\mathbb{Y}}}
\newcommand{\bbZ}{{\mathbb{Z}}}
\newcommand{\bbone}{{\mathbbm{1}}}

\newcommand{\bfA}{{\mathbf{A}}}
\newcommand{\bfB}{{\mathbf{B}}}
\newcommand{\bfC}{{\mathbf{C}}}
\newcommand{\bfD}{{\mathbf{D}}}
\newcommand{\bfE}{{\mathbf{E}}}
\newcommand{\bfF}{{\mathbf{F}}}
\newcommand{\bfG}{{\mathbf{G}}}
\newcommand{\bfH}{{\mathbf{H}}}
\newcommand{\bfI}{{\mathbf{I}}}
\newcommand{\bfJ}{{\mathbf{J}}}
\newcommand{\bfK}{{\mathbf{K}}}
\newcommand{\bfL}{{\mathbf{L}}}
\newcommand{\bfM}{{\mathbf{M}}}
\newcommand{\bfN}{{\mathbf{N}}}
\newcommand{\bfO}{{\mathbf{O}}}
\newcommand{\bfP}{{\mathbf{P}}}
\newcommand{\bfQ}{{\mathbf{Q}}}
\newcommand{\bfR}{{\mathbf{R}}}
\newcommand{\bfS}{{\mathbf{S}}}
\newcommand{\bfT}{{\mathbf{T}}}
\newcommand{\bfU}{{\mathbf{U}}}
\newcommand{\bfV}{{\mathbf{V}}}
\newcommand{\bfW}{{\mathbf{W}}}
\newcommand{\bfX}{{\mathbf{X}}}
\newcommand{\bfY}{{\mathbf{Y}}}
\newcommand{\bfZ}{{\mathbf{Z}}}
\newcommand{\bfa}{{\mathbf{a}}}
\newcommand{\bfb}{{\mathbf{b}}}
\newcommand{\bfc}{{\mathbf{c}}}
\newcommand{\bfd}{{\mathbf{d}}}
\newcommand{\bfe}{{\mathbf{e}}}
\newcommand{\bff}{{\mathbf{f}}}
\newcommand{\bfg}{{\mathbf{g}}}
\newcommand{\bfh}{{\mathbf{h}}}
\newcommand{\bfi}{{\mathbf{i}}}
\newcommand{\bfj}{{\mathbf{j}}}
\newcommand{\bfk}{{\mathbf{k}}}
\newcommand{\bfl}{{\mathbf{l}}}
\newcommand{\bfm}{{\mathbf{m}}}
\newcommand{\bfn}{{\mathbf{n}}}
\newcommand{\bfo}{{\mathbf{o}}}
\newcommand{\bfp}{{\mathbf{p}}}
\newcommand{\bfq}{{\mathbf{q}}}
\newcommand{\bfr}{{\mathbf{r}}}
\newcommand{\bfs}{{\mathbf{s}}}
\newcommand{\bft}{{\mathbf{t}}}
\newcommand{\bfu}{{\mathbf{u}}}
\newcommand{\bfv}{{\mathbf{v}}}
\newcommand{\bfw}{{\mathbf{w}}}
\newcommand{\bfx}{{\mathbf{x}}}
\newcommand{\bfy}{{\mathbf{y}}}
\newcommand{\bfz}{{\mathbf{z}}}
\newcommand{\bfzero}{{\mathbf{0}}}
\newcommand{\bfone}{{\mathbf{1}}}
\newcommand{\bfGamma}{{\boldsymbol \Gamma}}
\newcommand{\bfDelta}{{\boldsymbol \Delta}}
\newcommand{\bfTheta}{{\boldsymbol \Theta}}
\newcommand{\bfLambda}{{\boldsymbol \Lambda}}
\newcommand{\bfXi}{{\boldsymbol \Xi}}
\newcommand{\bfPi}{{\boldsymbol \Pi}}
\newcommand{\bfSigma}{{\boldsymbol \Sigma}}
\newcommand{\bfUpsilon}{{\boldsymbol \Upsilon}}
\newcommand{\bfPhi}{{\boldsymbol \Phi}}
\newcommand{\bfPsi}{{\boldsymbol \Psi}}
\newcommand{\bfOmega}{{\boldsymbol \Omega}}
\newcommand{\bfalpha}{{\boldsymbol \alpha}}
\newcommand{\bfbeta}{{\boldsymbol \beta}}
\newcommand{\bfgamma}{{\boldsymbol \gamma}}
\newcommand{\bfdelta}{{\boldsymbol \delta}}
\newcommand{\bfeps}{{\boldsymbol \eps}}
\newcommand{\bfzeta}{{\boldsymbol \zeta}}
\newcommand{\bfeta}{{\boldsymbol \eta}}
\newcommand{\bftheta}{{\boldsymbol \theta}}
\newcommand{\bfiota}{{\boldsymbol \iota}}
\newcommand{\bfkappa}{{\boldsymbol \kappa}}
\newcommand{\bflambda}{{\boldsymbol \lambda}}
\newcommand{\bfmu}{{\boldsymbol \mu}}
\newcommand{\bfnu}{{\boldsymbol \nu}}
\newcommand{\bfxi}{{\boldsymbol \xi}}
\newcommand{\bfpi}{{\boldsymbol \pi}}
\newcommand{\bfrho}{{\boldsymbol \rho}}
\newcommand{\bfsigma}{{\boldsymbol \sigma}}
\newcommand{\bftau}{{\boldsymbol \tau}}
\newcommand{\bfupsilon}{{\boldsymbol \upsilon}}
\newcommand{\bfphi}{{\boldsymbol \phi}}
\newcommand{\bfchi}{{\boldsymbol \chi}}
\newcommand{\bfpsi}{{\boldsymbol \psi}}
\newcommand{\bfomega}{{\boldsymbol \omega}}

\newcommand{\calA}{{\mathcal{A}}}
\newcommand{\calB}{{\mathcal{B}}}
\newcommand{\calC}{{\mathcal{C}}}
\newcommand{\calD}{{\mathcal{D}}}
\newcommand{\calE}{{\mathcal{E}}}
\newcommand{\calF}{{\mathcal{F}}}
\newcommand{\calG}{{\mathcal{G}}}
\newcommand{\calH}{{\mathcal{H}}}
\newcommand{\calI}{{\mathcal{I}}}
\newcommand{\calJ}{{\mathcal{J}}}
\newcommand{\calK}{{\mathcal{K}}}
\newcommand{\calL}{{\mathcal{L}}}
\newcommand{\calM}{{\mathcal{M}}}
\newcommand{\calN}{{\mathcal{N}}}
\newcommand{\calO}{{\mathcal{O}}}
\newcommand{\calP}{{\mathcal{P}}}
\newcommand{\calQ}{{\mathcal{Q}}}
\newcommand{\calR}{{\mathcal{R}}}
\newcommand{\calS}{{\mathcal{S}}}
\newcommand{\calT}{{\mathcal{T}}}
\newcommand{\calU}{{\mathcal{U}}}
\newcommand{\calV}{{\mathcal{V}}}
\newcommand{\calW}{{\mathcal{W}}}
\newcommand{\calX}{{\mathcal{X}}}
\newcommand{\calY}{{\mathcal{Y}}}
\newcommand{\calZ}{{\mathcal{Z}}}

\DeclareMathOperator{\groupGF}{GF}
\DeclareMathOperator{\groupGL}{GL}
\DeclareMathOperator{\groupSL}{SL}
\DeclareMathOperator{\groupO}{O}
\DeclareMathOperator{\groupSO}{SO}
\DeclareMathOperator{\groupU}{U}
\DeclareMathOperator{\groupSU}{SU}
\DeclareMathOperator{\groupE}{E}
\DeclareMathOperator{\groupSE}{SE}


\graphicspath{{fig/}}



\theoremstyle{remark}
\newtheorem*{remark}{Remark}
\newtheorem{lem}{Lemma}
\newtheorem{defn}{Definition}

% \newenvironment{sol}{
%   %\captionsetup[figure]{font={color=black}}
%   \textbf{Solution:}\\
%   \begingroup \color{red}
% }{
%   \endgroup
%   \qed
% }
\NewEnviron{sol}{
  \ifshowsol
  %\captionsetup[figure]{font={color=black}}
  \textbf{Solution:}\\
  \begingroup \color{black}
  \expandafter
  \BODY
  \endgroup
  \qed 
  \fi
}

\NewEnviron{Sol}{
  \ifshowsol
  %\captionsetup[figure]{font={color=black}}
  \textbf{Solution:}\\
  \begingroup \color{black}
  \expandafter
  \BODY
  \endgroup
  \qed 
  \fi
}

\let\vec=\boldsymbol
%%%%%%%%%%%%%%%%%% style.sty ends %%%%%%%%%%%%%%%%%%%%%%%%%%
\newtheorem{proposition}{proposition}[section]
\theoremstyle{remark}
\newtheorem*{hint}{Hint}
\newif\ifshowsol
\showsoltrue
%\showsolfalse
\title{
Predictive Machine Learning \ CS395T(53155) CSE392(65519) M393C(55759)\\ Fall 2022 TTH 3:30p.m.-5:00p.m. \ GDC 6.202 \\}
\author{Assignment 1}
\date{August 24, 2022} 


\begin{document}

\maketitle
\section*{Note:}
\begin{itemize}
\item Due 11:59pm September 7, 2022 
\item There are 5 problems to solve, and the last problem contains extra credit parts.
\item Students may discuss the homework problem sets with one another, but
{\bf solutions must be written by yourself}. Copying homework solutions from
another student, from a textbook, from a webpage, or from any other source is
not allowed.
\item You should make a statement on the persons with whom the exercise solution
was discussed and the material that was consulted to help you solve the
exercise. All solutions should be written in your own words.
\item Please typeset your solutions if possible. If you submit handwritten solutions they must be written legibly. \item \textbf{The solution must be turned in via the Canvas assignment page}. See instructions \href{https://community.canvaslms.com/docs/DOC-10701#jive_content_id_Assignments}{here} if you are a first-time user. A solution Latex template is also provided. However, you can use any Word processor. Legibly handwritten and scanned submitted solutions are also acceptable. 


\end{itemize}

\newpage

\section*{Exercise 1 [20pts]}

We are going to explore some basic properties for skew matrices ($A=-A^T$, if $A$ is a matrix). The discussion is specialized to 2D and 3D domains. 

\begin{description}
\item[(1)][5pts] Prove that any orthogonal matrix in $M_2(\mathbb{R})$ is either a rotation $R_\theta$ about the origin with angle of rotation $\theta$ or a reflection $\rho_\theta$ about the line passing through origin making an angle $\theta/2$. where

\begin{equation*}
R_\theta = \begin{pmatrix}
\cos \theta &amp; -\sin \theta \\ \sin \theta &amp; \cos \theta 
\end{pmatrix},
\rho_\theta = \begin{pmatrix}
\cos \theta &amp; \sin \theta \\ \sin \theta &amp; -\cos \theta 
\end{pmatrix}
\end{equation*}

\item[(2)][5pts] Given real numbers $x, y, z$, find (real and complex) eigenvalues for
$$
\begin{bmatrix}
0 &amp; -z &amp; y\\
z &amp; 0 &amp; -x \\
-y &amp; x &amp; 0
\end{bmatrix}
$$

\item[(3)][10pts] The exponential map for a square matrix $A$ is given by $$
\exp(A):= \sum\limits_{k=0}^{\infty}\frac{1}{k!}A^k.
$$
Derive an explicit expression for
$$
\exp\Big(
\begin{bmatrix}
0 &amp; -z &amp; y\\
z &amp; 0 &amp; -x \\
-y &amp; x &amp; 0
\end{bmatrix}
\Big).
$$
and simplify it.
\vspace{0.1in}
(Hint: Apply Eigen decomposition, but you do not have to compute all eigenvectors explicitly. Euler's formula might be helpful: $e^{i\theta} = cos \theta + i\sin \theta$)
\end{description}
\begin{sol}

\end{sol}

\section*{Exercise 2 [20pts]}

\begin{description}
\item[(1)][10pts] This exercise derives the normalizing constant of p.d.f. of a 1D random variable with Gaussian distribution. The normalizing constant for a zero-mean Gaussian is given by:
\begin{equation*}
    Z:= \int_{-\infty}^{\infty} \exp(-\frac{x^2}{2\sigma^2}) dx.
\end{equation*}
This integral is not an easy one at first glance. The trick is to compute:
$$Z^2 = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty}\exp(-\frac{x^2+y^2}{2\sigma^2}) dxdy $$
instead using polar coordinate transformation. Compute the integral of $Z^2$ and conclude that $Z=\sigma\sqrt{2\pi}$.
\item[(2)][10pts] Show the following relations between mutual information $I(X;Y)$ and individual and conditional entropies $H(X),H(X|Y)$, respectively.
\begin{equation*}
    I(X;Y) = H(X) - H(X|Y)=  H(Y)- H(Y|X),
\end{equation*}

where
\[ I(X;Y):=\sum_{x}\sum_{y}p(x,y)\log\frac{p(x,y)}{p(x)p(y)}.\]
\[ H(X):=-\sum_{x}p(x)\log p(x).\]
\[ H(X|Y):=-\sum_{y}p(y)\sum_{x}p(x|y)\log p(x|y).\]
\end{description}
\begin{sol}

\end{sol}

\section*{Exercise 3 [20pts]}

This exercise helps review some basic concept in probability distribution and some useful inequalities. \href{https://utexas.instructure.com/courses/1339687/files/67243777}{Class note 3} covers  some of the those concepts. For different probability inequalities, one can consult [BHK] Chapter 12.

\begin{description}
\item[(1)][5pts] Given the distribution $\frac{1}{3 \sqrt{2 \pi}}e^{-x^2 / 18}$,
what is the probability that $x &gt; 2$? Express your answer in terms of the error
function $\mathrm{erf}(\xi) = \frac{2}{\sqrt{\pi}} \int_0^\xi e^{-t^2}
\, \operatorname{d} t$.

\item[(2)][5pts]
For each of the following probability distributions, apply the Markov
inequality and Chebyshev inequality to bound $\Pr [x \geq a]$, where $a &gt; 1$.
For what value(s) of $a$ is the Chebyshev bound tighter in each case?
\\
a, $p(x) = 
\begin{cases}
1 , &amp; x = 1\\
0 , &amp; \text{otherwise}
\end{cases}$


b, $ p(x) = 
\begin{cases}
\frac{1}{2} , &amp; 0 \leq x \leq 2\\
0 , &amp; \text{otherwise}
\end{cases}$

\item[(3)][10pts] Let $s$ be the sum of $n$ independent random variables $x_1, x_2, . . . , x_n$, where for each $i$, \\
\[ x_i = 
\begin{cases}
0 &amp; \text{with probability }  p \\
1 &amp; \text{with probability }  1 - p
\end{cases}.
\]
Let $m = E(s)$. Show that $\Pr [ |s - m| &gt; 0.5 m] \leq 2 (8 e / 27)^{m/2}$.

(Hint: Use Chernoff bound)
\end{description}
\begin{sol}

\end{sol}

\section*{Exercise 4 [20pts]}

Given linear classifier (probably with noise $\epsilon$) $y=\vec{x}^T\vec{\beta}  + \epsilon,\epsilon\sim \mathcal{N}(0,\sigma^2)$. Assuming we collect data-observation pair $(\vec{x}_i,y_i),~i=1,2,\ldots,N,$ and consider the matrix form:
$$\vec{y}=X\vec{\beta} + \vec{e} \quad \vec{y}\in\bbR^N,\quad X\in\bbR^{N\times d}\quad \vec{e}\in\mathbb{R}^{N}.$$
Here $\vec{e}$ is the noise, whose components are i.i.d. Gaussian. You can always assume this is an over constrained system, i.e. $N&gt;d$. We have shown in the class the ridge regression. 
\begin{description}
\item[(1)][10pts] Using SVD decomposition, find the solution of least square optimization problem:
\begin{equation}
    \hat{\vec{\beta}}:= \arg\min_{\vec{\beta}}\, \|\vec{y}-X\vec{\beta}\|_2^2
\end{equation}
 Further, show that $f(\vec{x}):=\vec{x}^T\hat{\vec{\beta}}$ is the unbiased estimator of $\vec{x}^T\vec{\beta}$ given $X$ fixed. The definition of an unbiased estimator can be found in \href{https://utexas.instructure.com/courses/1339687/files/67243777}{Class note 3}.
\item[(2)][10pts] Prove the \textit{Gauss-Markov Theorem}: If $X$ are fixed, then the least square estimator of the classifier $f(\vec{x})= \vec{x}^T\hat{\vec{\beta}}$ has variance no bigger than that any of linear unbiased estimator. That is, for every $\vec{x}\in\bbR^N$, if $\exists~\vec{c} \in\bbR^N$ such that $\bbE[\vec{c}^T\vec{y}]=\bbE[\vec{x}^T\vec{\beta}]$, then $$\Var[\vec{x}^T\hat{\vec{\beta}}]\leq \Var[\vec{c}^T\vec{y}].$$
\end{description}
\begin{sol}

\end{sol}

\section*{Exercise 5 [20pts+extra]}

Consider the following continuous time linear dynamical system (CT LDS) with the following form:

\begin{equation*}
\dot{x}(t) = A(t)x(t) + B(t)u(t),\quad y(t)=C(t)x(t),\quad x(0)=0.
\end{equation*}

In conventional setting, $x(t)$ refers to the state (vector), $u(t)$ refers to the input or control, and $y(t)$ is the observation or output. The matrix $A(t)$ refers to the dynamics matrix, $B(t)$ refers to the input matrix and $C(t)$ refers to the output or sensor matrix.

In this problem, we only consider a \textit{time-invariant} dynamical system setup, where $A(t),B(t),C(t)$ are time independent and the model becomes:
\begin{equation*}
\dot{x}(t) = Ax(t) + Bu(t),\quad y(t)=Cx(t),\quad x(0)=0.
\end{equation*}

\begin{description}
\item[(1)][10pts] Assume $x(t)\in\mathbb{R}^{n},u(t)\in\mathbb{R}^{m},y(t)\in\mathbb{R}^{m}, A\in \mathbb{R}^{n\times n}, B\in\mathbb{R}^{n\times m},C\in\mathbb{R}^{m\times n}$, find a \textbf{constant} control function $u(t):\mathbb{R}_{+}\rightarrow \mathbb{R}^m$ such that $\lim_{t\rightarrow \infty} y(t) = y_0$ becomes a fixed time independent observation as the system evolves? You can assume $A,B,C$ are sufficiently good matrix so that the system is stable and can converge to a fixed point. (Hint: what is the easiest case that the linear dynamical system stalls at a fixed point?)

\item[(2)][10pts] Let $C=\mathbf{I}$, the identity matrix and define $u(t)=Kx(t)$, where $K\in\mathbb{R}^{m\times n}$ is the controller (matrix). One concerns whether or not we can find an appropriate controller such that the eigenvalues of matrix $A+BK$ can be the assigned value (and henceforth the dynamic is known to be stable). Consider the following dynamics of a scalar state $x(t)$:

\begin{equation}
    \dot{x}(t)=\begin{bmatrix}
    2 &amp; -3 &amp; 1  \\
    3 &amp; 2  &amp; 2 \\
    1 &amp; 3 &amp; 2 \\ 
    \end{bmatrix}x(t)+  \begin{bmatrix}
    1\\ -1 \\ 3 
    \end{bmatrix}u(t), \quad x(0)=x_0.
    \label{eq:1}
\end{equation}

Find $K=[k_1,k_2,k_3]$, such that $A+BK$ has eigenvalues $\lambda_1=-1,\lambda_2= -2,\lambda_3= -3$. 

\item[(3)][20pts, \textbf{extra}] Plot the dynamics stated in \eqref{eq:1} with $K$ you have computed and $x_0=[1,1,1]$. Note that your plot shall be a 3D curve. Further, plot three decomposed \textit{modes} of the dynamics:
\[ \dot{x}_i(t) := \dot{c}_i(t) u_i = c_i(t)\lambda_i u_i,\quad i=1,2,3.\]

Here $c_i(t)$ is a scalar function with respect to time and $u_i$ is the corresponding eigenvector of $\lambda_i$.
What do you think of the relationship between the overall dynamics and those decomposed dynamics? How would you determine $c_i(0)$? \textbf{You need to submit your code as a runnable python script or jupyter notebook and reproduce the result you have in order to obtain full credit.}

\begin{remark}
In this assignment you are required to write  codes in Python to plot the result. If you are new to python, here are some links as additional help resources:
\begin{itemize}
\item \href{https://www.anaconda.com/download/}{Anaconda Python Environment}
\item \href{https://docs.python.org/3/tutorial/}{Python tutorial (latest version)}
\item \href{http://jupyter.org/}{Jupyter Notebook}
\item Related Useful Packages: 
\href{https://www.numpy.org/}{Numpy}
\href{https://www.scipy.org/}{Scipy}
\href{https://matplotlib.org/}{Matplotlib}
\item \textbf{Important}:\href{https://scipy-cookbook.readthedocs.io/items/CoupledSpringMassSystem.html}{One example of solving an ODE system.}
\end{itemize}
\end{remark}
\end{description}
\begin{sol}

\end{sol}

\end{document}</pre></body></html>