\documentclass[fleqn,11pt]{article}
\usepackage{amsmath,amsfonts,amssymb,graphicx}
\usepackage{wrapfig}
\def\Ebox#1#2{%
\begin{center}
\includegraphics[width=#1\hsize]{#2}
\end{center}
}
\newcounter{arabnum}
\newenvironment{arabnum}{\begin{list}{{\upshape \arabic{arabnum}.\ }}{\usecounter{arabnum}
\setlength{\leftmargin}{24pt}
\setlength{\rightmargin}{8pt}
\setlength{\itemindent}{-1pt}
}}{\end{list}}
\newcounter{rmnum}
\newenvironment{romannum}{\begin{list}{{\upshape (\roman{rmnum})}}{\usecounter{rmnum}
\setlength{\leftmargin}{24pt}
\setlength{\rightmargin}{8pt}
\setlength{\itemindent}{-1pt}
}}{\end{list}}
\newcounter{anum}
\newenvironment{alphanum}{\begin{list}{{\upshape (\alph{anum})}}{\usecounter{anum}
\setlength{\leftmargin}{24pt}
\setlength{\rightmargin}{8pt}
\setlength{\itemindent}{-1pt}
}}{\end{list}}
\setlength{\parindent}{0in}
\pagestyle{empty}
\setlength{\textheight}{8.5in}
\setlength{\textwidth}{6in}
\setlength{\parsep}{.15in}
\setlength{\oddsidemargin}{.1in}
\setlength{\topmargin}{-.5cm}
\def\subhead#1{\smallskip\subparagraph{#1}}
\def\head#1{\subsection*{#1}}
\def\R{{\mathbb R}}
\begin{document}
\null\vspace{-1.75cm}
{\Large
{\bf ECE 515 / ME 540 \hfill Assignment \# 2}
\vspace{.15in}
Issued: Aug 31 \hfill Due: Sep 7, 2023}
\rule{6in}{.01in}
\vspace*{.1cm}
\head{Reading Assignment:}
\textbf{BMP}, Ch. 2.
\head{Problems:}
\begin{arabnum}
\setcounter{arabnum}{0}
\item (Exercise 2.11.1 in BMP) Which of the following sets are fields? Justify your answers.
\begin{romannum}
\item The set of integers with the usual operations.
\item The set of rational numbers with the usual operations.
\item The set of polynomials of degree less than $3$ with real coefficients with addition and multiplication of polynomials.
\item The set of all $n \times n$ nonsingular matrices with real-valued entries.
\item The set \{0, 1\} with addition being binary ``exclusive-or'' and multiplication being binary ``and''.
\end{romannum}
\item Which of the following are vector spaces over $\R$ (with
respect to the standard addition and scalar multiplication)?
Justify your answers.
\begin{romannum}
\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
$\frac{q(s)}{p(s)}$, where $q$ and $p$ are polynomials in
the complex variable $s$
and the degree of
$p$ does not exceed a given fixed positive integer $k$.
\item The space $L^2(\R)$ of square-integrable
functions, i.e., functions $f:\R\to\R$ with the property that
$\int_{-\infty}^{\infty}f^2(t)dt<\infty$.
\end{romannum}
\item
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 $\R^2$.
\item Let $A$ be the linear operator from the previous problem. Compute the matrix of $A$ relative to the basis
$ \bigg\{\begin{pmatrix}
-1 \\
0
\end{pmatrix},\begin{pmatrix}
2 \\
1
\end{pmatrix}\bigg\}.
$
\item Let $A:V\to W$ be a linear operator between finite-dimensional vector spaces.
\begin{romannum}
\item
Prove that $\dim N(A)+\dim R(A)=\dim V$ (the sum
of the dimension of the nullspace of $A$ and the dimension of the
range of $A$ equals the dimension of $V$).
\item Now assume that $V=W$. It is \textit{not} always true that $V$ is
a direct sum of $N(A)$ and $R(A)$. Find a counterexample
demonstrating this. Also, describe a class of linear operators
(as general as you can think of) for which this statement \textit{is} true.
\smallskip
\textit{Reminder:} Let $W_1$ and $W_2$ be subspaces of a vector space $V$. We say that $V$ is a \textit{direct sum} of $W_1$ and $W_2$, and write $V = W_1 \oplus W_2$, if $V = W_1 + W_2$ (i.e., any vector $v \in V$ can be represented as a sum $w_1 + w_2$ with $w_i \in W_i$, $i=1,2$) and $W_1 \cap W_2 = \{0\}$.
\end{romannum}
\end{arabnum}
\end{document}