\subsection{Due: 02/01}

\subsubsection{Problem 1}

Convert each of the following higher-order differential equations into
state-space form:

\begin{itemize}
\tightlist
\item
  System (a): \[\ddot x = \dot u + 3 u\]
\item
  System (b): \[\dfrac{d^3 x}{dt^3} - 2 \dfrac{d^2x}{dt^2} + x = u\]
\item
  System (c): \[\ddot x = \sin \left( \dot x - x \right)\]
\end{itemize}

\subsubsection{Problem 2}

Find the transfer function for the following systems:

\begin{itemize}
\item
  System (I): \[
   \begin{bmatrix}
   \dot x_1 \\ \dot x_2 
   \end{bmatrix} = \begin{bmatrix}  
   0 & -a_1 \\ 1 & -a_2 
   \end{bmatrix} + \begin{bmatrix} b_0 \\ b_1 \end{bmatrix} u; \qquad y = \begin{bmatrix}
   0 & 1
   \end{bmatrix}\begin{bmatrix}x_1 \\ x_2\end{bmatrix}
   \]
\item
  System (II): \[
   \begin{bmatrix}
   \dot x_1 \\ \dot x_2 
   \end{bmatrix} = \begin{bmatrix}  
   0 & 1 \\ -a_1 & -a_2 
   \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \end{bmatrix} u; \qquad y = \begin{bmatrix}
   b_0 & b_1
   \end{bmatrix}\begin{bmatrix}x_1 \\ x_2\end{bmatrix}
   \]
\end{itemize}

\subsubsection{Problem 3}

Let \(A\) be the linear operator in the plane corresponding to the
counter-clockwise rotation around the origin by some given angle
\(\theta\). See \href{hw01.html\#problem-2}{Problem 2} in Homework 1.
Compute the matrix of \(A\) relative to the basis:

\[
\left\{ \begin{bmatrix} -1 \\ 0 \end{bmatrix}, \begin{bmatrix} 2 \\ 1 \end{bmatrix}  \right\}
\]

\subsubsection{Problem 4}

Let us revisit \href{hw01.html\#problem-5}{Problem 5} from Homework 1.
Alice and Bob have reason to believe that Cheng's performance scores and
bonus amount that he gave them were incorrect. They still want to figure
out their unknown weights, but now they have to use only their own
scores and bonus amounts, without Cheng's.

\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\item
  Set up this problem as solving a linear equation of the form
  \(Ax = b\) for the unknown vector \(x\). The new matrix \(A\) will be
  \(2 \times 3\)
\item
  Let \(A^\dagger\) be a \emph{right inverse} for \(A\), which by
  definition is a matrix such that \(AA^\dagger=I\). Express all
  solutions of \(x\) in the equation in (a) above in terms of such a
  right inverse \(A^\dagger\), the vector \(b\) and the nullspace of
  \(A\). Carefully state and justify your answer.
\item
  Find a particular solution \(x^*\) of the equation in (a).
  \textbf{Hint:} Use your solution from Homework 1. Verify numerically
  that \(x^∗\) indeed belongs to the solution space you described in (b)
  using some CAS.
\end{enumerate}

\subsubsection{Problem 5}

Let \(A\) be a real valued \(n \times n\) matrix. Suppose that
\(\lambda + i \mu\) is a complex eigenvalue of \(A\) and \(x + iy\) is a
corresponding complex eigenvector. Here \(\lambda, \mu \in \mathbb{R}\)
and \(x, y \in \mathbb{R}^n\).

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\item
  Show that \(x-iy\) is also an eigenvector with eigenvalue
  \(\lambda - i \mu\).
\item
  Let \(V\) be the 2-dimensional subspace over \(\mathbb{R}\) spanned by
  \(x\) and \(y\). In other words, \(V\) is the set of linear
  combinations, with real coefficients, of the real-valued vectors \(x\)
  and \(y\). Show that \(V\) is an invariant subspace for \(A\), in the
  sense that for every \(z \in V\) we have \(Az  \in V\).
\end{enumerate}

\subsubsection{Problem 6}

Consider the following matrix, whose exponential we derive in class:

\[
A = \begin{bmatrix} a & b \\ -b & a \end{bmatrix} \qquad a, b \in \mathbb{R}
\]

\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
  Re-derive an expression for \(e^{At}\) by diagonalizing \(A\) over
  \(\mathbb{C}\) and using the formula
  \[e^{a \pm ib}:= e^a \cos b \pm ie^a \sin b\] (treat this as the
  \emph{definition} of the complex exponential).
\item
  Using the formila \(e^{A(t+\sigma)} = e^{At}e^{A \sigma}\) and the
  result of (a), for suitably chosen values of \(a\) and \(b\), verify
  the trigonometric identities: \[
    \begin{align}
    \cos (t+\sigma) &= \cos t \cos \sigma - \sin t \sin \sigma \\
    \sin (t + \sigma) &= \sin t \cos \sigma + \cos t \sin \sigma
    \end{align}
    \]
\end{enumerate}
