\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig, amsmath, latexsym}
\usepackage[all]{xy}
\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in

\newcommand{\C}{\mathbb{C}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\DeclareMathOperator{\im}{Im}
\DeclareMathOperator{\Span}{span}
\DeclareMathOperator{\Char}{char}
\DeclareMathOperator{\rank}{rank}

\renewcommand{\footrulewidth}{1pt}
\newenvironment{proof}[1][Proof.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{sol}[1][Solution.]{\smallskip\begin{trivlist}
    \item[\hskip \labelsep {\sffamily #1}]}{\qed\end{trivlist}\bigskip}
\newenvironment{fminipage}%
{\setlength{\fboxsep}{15pt}\begin{Sbox}\begin{minipage}}%
{\end{minipage}\end{Sbox}\fbox{\TheSbox}}
\newcommand{\qed}{\hfill \ensuremath{\Box}}
\newcommand{\disp}{\displaystyle}


\pagestyle{fancy} \lhead{{\sf Homework 4 $|$ Due February 10
    (Tuesday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Spring 2015 $|$ M.~Macauley}}


\begin{document}

$\;$

Throughout, $X$ is assumed to be a vector space of dimension
$n<\infty$.

\begin{enumerate}

% Lax, Exercise 3.7
\item Show that whenever meaningful, 
  \begin{enumerate}
  \item $(ST)'=T'S'$
  \item $(T+R)'=T'+R'$
  \item $(T^{-1})'=(T')^{-1}$.
  \end{enumerate}
  Here, $S'$ denotes the transpose of $S$. Carefully describe what you
  mean by ``whenever meaningful'' in each case. \\
  


  % Proof of Theorem 5' from Lax
\item Give a direct algebraic proof of
  $N_{T'}^\perp=(R_T^\perp)^\perp$. (That is, don't just use the fact
  that $N_{T'}=R_T^\perp$ and take the annihilator of both sides.) \\



  % Lax Exercise 3.10, 3.11
\item Let $A,B\colon X\to X$ be linear maps.
  \begin{enumerate} 
  \item Show that if $A$ is invertible and similar to $B$, then $B$ is
    also invertible, and $B^{-1}$ is similar to $A^{-1}$.
  \item Show that if either $A$ or $B$ is invertible, then $AB$ and
    $BA$ are similar. \\
  \end{enumerate}



  % Lax Exercise 3.14
\item Suppose $T\colon X\to X$ is a linear map of rank 1.
  \begin{enumerate}
  \item Show that there exists $c\in K$ such that $T^2=cT$.
  \item Show that if $c\neq 1$, then $I-T$ has an inverse. \\
  \end{enumerate}



% Lax Exercise 3.15
  \item Suppose that $S,T\colon X\to X$ are linear maps.
  \begin{enumerate}
    \item Show that $\rank(S+T)\leq\rank(S)+\rank(T)$.
    \item Show that $\rank(ST)\leq\rank(S)$.
    \item Show that $\dim(N_{ST})\leq\dim N_S+\dim N_T$.
  \end{enumerate}
  For each of these, give an explicit example showing how equality
  need not hold. \\



\item Let $T:X\to X$ be linear, with $\dim X=n$.
  \begin{enumerate}
  \item Prove that if $T^2=T$, then $X=R_T\oplus N_T$. 
  \item Show by example that if $T^2\neq T$, then $X=R_T\oplus N_T$
    need not hold.
  \item Prove that $N_{T^n}=N_{T^{n+1}}$ and
    $R_{T^n}=R_{T^{n+1}}$. 
  \item Prove that $X=R_{T^n}\oplus N_{T^n}$.
  \item Show there exists a linear map $S:X\to X$ such that $ST=TS$
    and $ST^{n+1}=T^n$.
  \end{enumerate}
  

\end{enumerate}

\end{document}
