\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig, amsmath, latexsym}
%\usepackage[all]{xy}
\usepackage{tikz}
\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}
\usetikzlibrary{decorations.pathreplacing}
\usetikzlibrary{patterns}
\usetikzlibrary{shapes.geometric}
\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 3 $|$ Due January 30
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Spring 2026 $|$ M.~Macauley}}

\begin{document}

$\;$ \vspace{-1mm}

\begin{enumerate}

%%------------------------------------------------------------------------
  
\item Let $T\colon X\to U$ be a linear map between vector spaces. Prove that the image of a subspace $Y\leq X$ is a subspace of $U$. \medskip

%%------------------------------------------------------------------------

\item Let $T\colon X\to U$ be a linear map between vector spaces. Prove that the inverse image of a subspace $V\leq U$ is a subspace of $X$. \medskip

%%------------------------------------------------------------------------

\item Let $X$ and $U$ be vector spaces, and suppose that $Y$ is a subspace of $X$. Let $Q\colon X\to X/Y$ be the canonical quotient map sending $x\stackrel{Q}{\longmapsto}\{x\}$, and let $T\colon X\to U$ be a linear map. State and prove a condition that is both necessary and sufficient for the existence of a unique linear map $S\colon X/Y\to U$ such that $T=S\circ Q$. When this happens, the map $T$ is said to \emph{factor through} the quotient space, as shown by the following commutative diagram:
  \[
  \begin{tikzpicture}[scale=3.5,>= stealth,
      every label/.style={font=\footnotesize}]
    \begin{scope}[shift={(0,0)}]
      \node (X) at (0,0) {$X$};
      \node (U) at (1,0) {$U$};
      \node (X/Y) at (.5,-.5) {$X/Y$};
      \draw[->] (X) -- (U) node[midway,above]{$T$};
      \draw[->] (X) -- (X/Y) node[midway,below left]{$Q$};
      \draw[->,dashed] (X/Y) -- (U) node[midway,below right]{$S$};
    \end{scope}
  \end{tikzpicture}
  \]
  
%%------------------------------------------------------------------------

\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. \bigskip
  \end{enumerate}

%%------------------------------------------------------------------------

\item Let $S,T\colon X\to X$ be linear maps. Show that $\rank(S+T)\leq\rank(S)+\rank(T)$. Without using matrices, give an explicit example of where equality holds, and one where it does not. \medskip

%%------------------------------------------------------------------------

\item Let $S,T\colon X\to X$ be linear. Show that $\rank(ST)\leq\rank(S)$. Without using matrices, give an explicit example of where equality holds, and one where it does not.~ \medskip

%%------------------------------------------------------------------------
  
\item Let $S,T\colon X\to X$ be linear. Show that $\dim(N_{ST})\leq\dim N_S+\dim N_T$. Without using matrices, give an explicit example of where equality holds, and one where it does not.~  \medskip
 
%%------------------------------------------------------------------------

\item Let $T:X\to X$ be linear, with $\dim X=n$. Prove that if $T^2=T$, then $X=R_T\oplus N_T$. Show by example that if $T^2\neq T$, then $X=R_T\oplus N_T$ need not hold. \medskip

%%------------------------------------------------------------------------

\item Let $T:X\to X$ be linear, with $\dim X=n$.
\begin{enumerate}
  \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}
