\subsection{Due: TBD}
\subsubsection{Problem 1}
Which of the following are vector spaces over \(\mathbb{R}\) (with
respect to standard addition and scalar multiplication). Justify your
answers.
\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
The set of real valued \(n \times n\) matrices with nonnegative
entries where \(n\) is a given positive integer.
\item
The set of rational functions of the form \(\dfrac{p(s)}{q(s)}\) where
\(p\) and \(q\) are polynomials in the complex variable \(s\) and the
degree of \(q\) does not exceed a given fixed positive integer \(k\).
\item
The space \(L^2\left(\mathbb{R}, \mathbb{R}\right)\) of
square-integrable functions, i.e., functions
\(f : \mathbb{R} \to \mathbb{R}\) with the property that
\end{enumerate}
\[
\int \limits _{-\infty} ^{\infty} f^2 (t) dt < \infty
\]
\subsubsection{Problem 2}
Let \(A\) be the linear operator in the plane corresponding to the
counter-clockwise rotation around the origin by some given angle
\(\theta\). Compute the matrix of \(A\) relative to the standard basis
in \(\mathbb{R}^2\).
\subsubsection{Problem 3}
Let \(A: X \to Y\) be a linear transformation.
\begin{enumerate}
\def\labelenumi{(\alph{enumi})}
\tightlist
\item
Prove that \(\dim N (A) + \dim R(A) = \dim X\) (the sum of the
dimension of the nullspace of \(A\) and the dimension of the range of
\(A\) equals the dimension of \(X\)).
\item
Now assume that \(X = Y\). It is not always true that \(X\) is a
direct sum of \(N(A)\) and \(R(A)\). Find a counterexample
demonstrating this. Also, describe a class of linear transformations
(as general as you can think of) for which this statement is true.
\end{enumerate}
\subsubsection{Problem 4}
Consider the standard RLC circuit, except now allow its characteristics
\(R, L\) and \(C\) to vary with time. Starting with the same non-dynamic
physical laws as in class (\(q = CV_c\) for the capacitor charge,
\(\varphi = LI\) for the inductor flux), derive a dynamical model of
this circuit. It should take the form:
\[ \dot{x} = A(t) x + B(t) u \]
\includegraphics[width=0.35\textwidth,height=\textheight]{./figures/hw1_fig1.png}
\subsubsection{Problem 5}
Three employees --- let's call them Alice, Bob, and Cheng --- received
their end-of-the-year bonuses which their boss calculated as a linear
combination of three performance scores: leadership, communication, and
work quality. The coefficients (weights) in this linear combination are
the same for all three employees, but the boss doesn't disclose them.
Alice knows that she got the score of 4 for leadership, 4 for
communication, and 5 for work quality. Bob's scores for the same
categories were 3, 5, and 4, and Cheng's scores were 5, 3, and 3. The
bonus amounts are \$18000 for Alice, \$16000 for Bob, and \$14000 for
Cheng. The employees are now curious to determine the unknown
coefficients (weights).
\begin{enumerate}
\def\labelenumi{\alph{enumi}.}
\tightlist
\item
Set up this problem as solving a linear equation of the form
\(Ax = b\) for the unknown vector \(x\).
\item
Calculate the unknown weights. It's up to you whether you use part (a)
for this or do it another way.
\item
Are the weights that you computed unique? Explain why or why not.
\end{enumerate}