\documentclass[11pt]{amsart}  

\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{eucal}
\usepackage{yfonts}
\usepackage[all]{xy}
\usepackage{amsxtra}
\usepackage{appendix} 
 


\usepackage[pdftex, breaklinks, linktocpage=true, bookmarksopen=true,bookmarksopenlevel=0,bookmarksnumbered=true]{hyperref}

\hypersetup{
pdfauthor = {\textcopyright\ Bryden Cais},
pdfcreator = {\LaTeX\ with package \flqq hyperref\frqq},
colorlinks = {true}
} 



   \topmargin=-0.75in
   \oddsidemargin=-0.15in
   \evensidemargin=-0.15in
   \textwidth=6.8in
   \textheight=9.75in

%\xymatrixcolsep{2.8pc}
\CompileMatrices
\entrymodifiers={+!!<0pt,\fontdimen22\textfont2>}
%\def\objectstyle{\displaystyle}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Introduce a nice script font for sheaves

\DeclareFontFamily{OT1}{rsfs}{}
\DeclareFontShape{OT1}{rsfs}{n}{it}{<-> rsfs10}{}
\DeclareMathAlphabet{\mathscr}{OT1}{rsfs}{n}{it}

\DeclareFontFamily{OT1}{pzc}{}
\DeclareFontShape{OT1}{pzc}{n}{it}{<->s*[2.2]pzc}{}
\DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{b}{sl}

%\DeclareFontFamily{OT1}{pzc}{}
%\DeclareFontShape{OT1}{pzc}{m}{it}{<-> s * [0.900] pzcmi7t}{}
%\DeclareMathAlphabet{\mathpzc}{OT1}{pzc}{m}{it}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% These commands allow one to make roman numerals
\makeatletter
\newcommand{\rmnum}[1]{\romannumeral #1}
\newcommand{\Rmnum}[1]{\expandafter\@slowromancap\romannumeral #1@}
\makeatother

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Math Operators
\DeclareMathOperator{\Ann}{Ann}
\DeclareMathOperator{\depth}{depth}
\DeclareMathOperator{\zar}{zar}
\DeclareMathOperator{\Cat}{Cat}
\DeclareMathOperator{\Cov}{Cov}
\DeclareMathOperator{\can}{can}
\DeclareMathOperator{\tot}{tot}
\DeclareMathOperator{\gr}{gr}
\DeclareMathOperator{\id}{id}
\DeclareMathOperator{\spc}{sp}
\DeclareMathOperator{\Frac}{Frac}
\DeclareMathOperator{\ord}{ord} 
\DeclareMathOperator{\nil}{nil}
\DeclareMathOperator{\bij}{bij}
\DeclareMathOperator{\Sym}{Sym} 
\DeclareMathOperator{\Stab}{Stab} 
%
%\DeclareMathOperator{\pr}{pr}
\newcommand*{\pr}{\rho}
%
\DeclareMathOperator{\Div}{Div} 
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\End}{End}
\DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\Gal}{Gal}
\DeclareMathOperator{\GL}{GL}
\DeclareMathOperator{\SL}{SL} 
\DeclareMathOperator{\PSL}{PSL}
\DeclareMathOperator{\PGL}{PGL}
\DeclareMathOperator{\Aut}{Aut}
\DeclareMathOperator{\Spec}{Spec}
\DeclareMathOperator{\Spf}{Spf}
\DeclareMathOperator{\Sp}{Sp}
\DeclareMathOperator{\Proj}{Proj}
\DeclareMathOperator{\coker}{coker}
\DeclareMathOperator{\et}{\acute{e}t}
\DeclareMathOperator{\rig}{rig}
\DeclareMathOperator{\Fr}{Fr}
\DeclareMathOperator{\dR}{dR}
\DeclareMathOperator{\Cris}{Cris}
\DeclareMathOperator{\cris}{cris}
\DeclareMathOperator{\MW}{MW}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Tr}{Tr}
\DeclareMathOperator{\an}{an}
\DeclareMathOperator{\Ner}{N\acute{e}r}
\DeclareMathOperator{\Pic}{Pic}
\DeclareMathOperator{\Alb}{Alb}
\DeclareMathOperator{\sm}{sm}
\DeclareMathOperator{\Extrig}{Extrig}
\DeclareMathOperator{\Lie}{Lie}
\DeclareMathOperator{\Inf}{Inf}
%\DeclareMathOperator{\BT}{BT}
\DeclareMathOperator{\reg}{reg}
\DeclareMathOperator{\Ab}{Ab}
\DeclareMathOperator{\tors}{tors}
\DeclareMathOperator{\free}{fr}
\DeclareMathOperator{\red}{red}
\DeclareMathOperator{\smooth}{sm}
\DeclareMathOperator{\Char}{char}
\DeclareMathOperator{\nr}{nr}
\DeclareMathOperator{\Mod}{Mod}
\DeclareMathOperator{\Fil}{Fil}
\DeclareMathOperator{\MF}{MF}
\DeclareMathOperator{\rk}{rk}
\DeclareMathOperator{\cork}{co-rk}
\DeclareMathOperator{\st}{st}
\DeclareMathOperator{\Rep}{Rep}
\DeclareMathOperator{\tor}{tor}
\DeclareMathOperator{\sep}{s}
\DeclareMathOperator{\perf}{p}
\DeclareMathOperator{\un}{un}
\DeclareMathOperator{\projdim}{projdim}
\DeclareMathOperator{\res}{res}
\DeclareMathOperator{\Def}{Def}
\DeclareMathOperator{\Ht}{ht}
\DeclareMathOperator{\GD}{GD}
\DeclareMathOperator{\Ig}{Ig}
\DeclareMathOperator{\len}{length}
\DeclareMathOperator{\cone}{cone}
\DeclareMathOperator{\supp}{supp}
\DeclareMathOperator{\im}{im}
\DeclareMathOperator{\rad}{rad}
\DeclareMathOperator{\Con}{Conn}
\DeclareMathOperator{\Irr}{Irr}
\DeclareMathOperator{\bal}{bal.}
\DeclareMathOperator{\Ell}{Ell}
\DeclareMathOperator{\Sets}{Sets}
\DeclareMathOperator{\univ}{univ}
\DeclareMathOperator{\MC}{M}
\DeclareMathOperator{\Rad}{Rad}
\DeclareMathOperator{\abs}{abs}
\DeclareMathOperator{\Cot}{Cot}
\DeclareMathOperator{\pdiv}{pdiv}
\DeclareMathOperator{\ModFI}{ModFI}
\DeclareMathOperator{\inv}{IF}
\DeclareMathOperator{\mult}{m}
\DeclareMathOperator{\loc}{ll}
\DeclareMathOperator{\Jac}{Jac}
\DeclareMathOperator{\Frob}{F}
\DeclareMathOperator{\Ver}{V}
\DeclareMathOperator{\proj}{proj}
\DeclareMathOperator{\Mat}{Mat}
\DeclareMathOperator{\Span}{span}
\DeclareMathOperator{\Gr}{Gr}
\DeclareMathOperator{\surj}{surj}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%New Commands		

%%% The usual rings %%%
\newcommand*{\R}{\ensuremath{\mathbf{R}}}   
\renewcommand*{\c}{\ensuremath{\mathbf{C}}}              
\newcommand*{\Z}{\ensuremath{\mathbf{Z}}}               
\newcommand*{\Q}{\ensuremath{\mathbf{Q}}}                           
\newcommand*{\N}{\ensuremath{\mathbf{N}}}             
\newcommand*{\Qbar}{\overline{\Q}}                             
\renewcommand*{\k}{\kappa}             
\newcommand*{\kbar}{\overline{k}}       
\newcommand*{\Kbar}{\overline{K}}    
          
%%% The usual schemes %%%
\renewcommand*{\P}{\ensuremath{\mathbf{P}}}           		
\newcommand*{\Gm}{\ensuremath{{\mathbf{G}_m}}}   
\newcommand*{\Ga}{\ensuremath{{\mathbf{G}_a}}}   

%%% Gothic letters (ideals) %%%
\newcommand*{\m}{\mathfrak{M}}                          
\newcommand*{\n}{\mathfrak{N}}                            
\newcommand*{\p}{\mathfrak{p}}                            
\newcommand*{\q}{\mathfrak{q}}                            
\renewcommand*{\r}{\mathfrak{r}}       
\newcommand*{\s}{\mathfrak{S}}

%%% Capital scripts (sheaves etc)
\newcommand*{\A}{\ensuremath{\mathcal{A}}}
\newcommand*{\B}{\mathcal{B}}
\newcommand*{\scrC}{\mathscr{C}}
\newcommand*{\C}{\mathbf{C}}
\newcommand*{\E}{\mathscr{E}}     
\newcommand*{\F}{\mathbf{F}}
\newcommand*{\scrF}{\mathscr{F}}
\newcommand*{\scrS}{\mathscr{S}}
\newcommand*{\scrW}{\mathscr{W}}
\newcommand*{\G}{\mathcal{G}}
\newcommand*{\scrG}{\mathscr{G}}                           
\newcommand*{\h}{\mathscr{H}}                               
\newcommand*{\I}{\mathscr{I}}                               
\newcommand*{\J}{\mathcal{J}}
\renewcommand*{\L}{\mathscr{L}}
\newcommand*{\scrM}{\mathscr{M}}
\newcommand*{\calM}{\mathcal{M}}
\renewcommand*{\O}{\mathscr{O}}                    
\newcommand*{\U}{\mathscr{U}} 
\newcommand*{\X}{\mathcal{X}}     
\newcommand*{\Y}{\mathcal{Y}}

%%% script capitals %%%
\newcommand*{\scrJ}{\mathscr{J}}                               
\newcommand*{\calL}{\mathcal{L}}                              
\newcommand*{\calS}{\mathcal{S}}                             
\newcommand*{\scrP}{\mathscr{P}}                             

%%% Various sheaves %%%
\newcommand*{\scrHom}{\mathscr{H}\mathit{om}}      
\newcommand*{\scrExtrig}{\mathscr{E}\mathit{xtrig}}	
\newcommand*{\scrExt}{\mathscr{E}\mathit{xt}}               
\newcommand*{\scrLie}{\mathscr{L}\mathit{ie}}      
\newcommand*{\scrD}{\mathscr{D}}          

%%% Misc %%%

\newcommand*{\barrho}{\overline{\rho}}                        
\newcommand*{\invlim}{\varprojlim}                               
\newcommand*{\tensorhat}{\widehat{\otimes}}             
\newcommand*{\D}{\ensuremath{\mathbf{D}}}
\newcommand*{\M}{\ensuremath{\mathbf{M}}}
\newcommand*{\calD}{\ensuremath{\mathcal{D}}}              
\renewcommand*{\H}{\ensuremath{\mathfrak{H}}}
\newcommand*{\z}{\zeta} 
\newcommand*{\dual}{\vee}
\renewcommand*{\qedsymbol}{\ensuremath{\blacksquare}}            % end of proof
\newcommand*{\eqdef}{\stackrel{\text{def}}{=}}     % definition
\newcommand*{\wa}[1]{{}^{\mathrm{w.a.}}\hspace{-0.4ex}{#1}}
\newcommand*{\mfun}{\underline{\M}}
\newcommand*{\Mfun}{\underline{\m}}
\newcommand*{\Dfun}{\underline{D}}
\newcommand*{\Gfun}{\underline{G}}
\newcommand*{\Dual}[1]{\widehat{#1}}
%\newcommand*{\R}{\ensuremath{\mathscr{R}}}              
\renewcommand*{\b}{\ensuremath{\mathrm{b}}}
\newcommand*{\bnd}{\ensuremath{\mathrm{bnd}}}
\newcommand*{\e}{\ensuremath{\mathbf{e}}}
\newcommand*{\Dst}{\ensuremath{{\underline{D}_{\st}}}}
\renewcommand*{\div}{\ensuremath{\mathrm{div}}}
\renewcommand*{\SS}{\ensuremath{\underline{\mathrm{ss}}}}
\renewcommand*{\u}[1]{\underline{#1}}
\renewcommand*{\o}[1]{\overline{#1}}
\newcommand*{\wh}[1]{\widehat{#1}}
\newcommand*{\nor}[1]{{#1}^{\mathrm{n}}}
\newcommand*{\T}{\ensuremath{\mathbf{T}}}
\newcommand*{\tens}{\mathop{\otimes}\limits}

%\newcommand*{\Win}{\ensuremath{\mathscr{W}\!\mathit{in}}}
%\newcommand*{\Win}{\textswab{Win}}
%\newcommand*{\BT}{\textswab{BT}}
\DeclareMathOperator{\Win}{Win}
\DeclareMathOperator{\BT}{BT}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Define environments

\theoremstyle{plain}
  \newtheorem{theorem}{Theorem}
  \newtheorem{proposition}[theorem]{Proposition}
  \newtheorem{lemma}[theorem]{Lemma}
  \newtheorem{corollary}[theorem]{Corollary}

\theoremstyle{definition}
  \newtheorem{definition}[theorem]{Definition}
  \newtheorem{notation}[theorem]{Notation}
  \newtheorem{hypotheses}[theorem]{Hypotheses}
  \newtheorem{question}[theorem]{Question}
    \newtheorem{problem}[theorem]{Problem}

\theoremstyle{remark}
  \newtheorem{example}[theorem]{Example}
  \newtheorem{remark}[theorem]{Remark}
  \newtheorem{remarks}[theorem]{Remarks}
  \newtheorem{exercise}[theorem]{Exercise}
  \newtheorem{warning}[theorem]{Warning}
  \newtheorem{convention}[theorem]{Convention}	
	
\include{header}


\numberwithin{theorem}{section}  
\numberwithin{equation}{section}



\newenvironment{dotlist}
   {
      \begin{list}
         {$\cdot$}
         {
            \setlength{\itemsep}{.5ex}
            \setlength{\parsep}{0ex}
            \setlength{\leftmargin}{1em}
            \setlength{\parskip}{0ex}
            \setlength{\topsep}{0ex}
         }
   }
   {
      \end{list}
   }
  




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%This command creates a footnote and an indicator mark in the margin pointing to problems, questions, or unresolved (small) issues
%To use type \edit{ insert text here }

\newcommand{\marginalfootnote}[1]{%
        \footnote{#1}
        \marginpar[\hfill{\sf\thefootnote}]{{\sf\thefootnote}}}
\newcommand{\edit}[1]{\marginalfootnote{#1}}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%This command creates a box marked ``To Do'' around text.
%To use type \todo{  insert text here  }.

\newcommand{\todo}[1]{\vspace{5 mm}\par \noindent
\marginpar{\textsc{ToDo}}
\framebox{\begin{minipage}[c]{0.95 \textwidth}
\tt #1 \end{minipage}}\vspace{5 mm}\par}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\begin{document}
\bibliographystyle{plain}     

\title{Linear Algebra Problems\\ {\rm University of Arizona Integration Workshop, August 2017}}
%\author{Bryden Cais}
\address{University of Arizona, Tucson}
\email{cais@math.arizona.edu}
\thanks{These problems borrow heavily from versions from previous years,
especially those written by Nick Rogers, Dinesh Thakur, and Doug Ulmer, as well as
old University of Arizona Algebra Qualifying Exams.}



%\subjclass[2000]{Primary:  Secondary: }
%\keywords{}
\date{\today}


%\begin{abstract}
%	
%\end{abstract}

\maketitle

\section{Lecture 1}\label{lec1}

\begin{problem}
	Prove that matrix multiplication is associative by relating this operation to 
	composition of linear maps (via choices of bases).
\end{problem}

\begin{problem}
	Let $V$ be a vector space and $S$ a basis of $V$.
	\begin{enumerate}
		\item Prove that $S^*\subseteq V^*$ is a linearly independent set.
		\item If $V$ is of finite dimension, prove that $S^*$ is a basis of $V^*$.
		\item Give an example to show that $S^*$ need not span $V^*$ in general.
	\end{enumerate}
\end{problem}



\begin{problem}
	Let $U_1,U_2$ be subspaces of a finite dimensional vector space $V$.  Prove that
	$$\dim(U_1+U_2) = \dim(U_1)+ \dim(U_2) - \dim(U_1\cap U_2).$$
\end{problem}


\begin{problem}
	Prove or disprove: For $U_1,U_2,U_3$ subspaces of
	a finite dimensional vector space $V$,
	\begin{multline*}
		\dim(U_1+U_2+U_3)= \dim(U_1)+\dim(U_2)+ \dim(U_3)\\
		-\dim(U_1\cap U_2) -\dim(U_1\cap U_3)-\dim(U_2\cap U_3)+\dim(U_1\cap U_2\cap U_3).
	\end{multline*}
\end{problem}

\begin{problem}
	A system of linear equations over an infinite field may have no solutions, 
	a unique solution, or infinitely many solutions.  Explain this behavior in terms
	of kernels of linear transformations.
\end{problem}

\begin{problem}
	Let $V,W$ be vector spaces of dimensions $n$ and $m$, respectively.
	Show that $\Hom(V,W)$ is naturally a vector space and that it has dimension $mn$.
\end{problem}

\begin{problem}
	Let $\{W_i\}_i$ be a collection of vector spaces over a field $k$, indexed by some set $I$,
	and set $V:=\bigoplus_i W_i$.  Express $V^*$ in terms of the $W_i^*$.  Do not assume $I$ is
	finite, nor that the $W_i$ are of finite dimension.  
\end{problem}

\begin{problem}
	Let $V$ be a finite dimensional vector space and $T\in \Hom(V,V)$
	a linear map.
	\begin{enumerate}
		\item Show that $T$ is nilpotent if and only if there is a basis of $V$ in which the
		matrix of $T$ is upper triangular with zeroes on the diagonal.  Deduce
		that if $T$ is nilpotent, than $T^{\dim(V)}=0$.
		
		\item Assuming that $k$ is algebraically closed,
		show that $T$ is semisimple if and only if there is a basis of $V$ in which
		the matrix of $T$ is diagonal.
		
		\item Give an example of $V$ and $T$ which is neither nilpotent nor semisimple.  
	\end{enumerate}
\end{problem}

\begin{problem}
	Let $V$ be a vector space of dimension $n$ over a field $k$, and let $T\in \Hom(V,V)$.  
	\begin{enumerate}
		\item Prove that there is a basis of $V$ in which the matrix of $T$ is upper triangular 
		if and only if there is a chain of $T$-stable subspaces
		$$0=V_0 \subset V_1\subset \cdots \subset V_n=V$$
		with $\dim(V_i)=i$.
		
		\item If $k=\C$, then prove that $T$ has an eigenvector.  Hint: for any nonzero $v\in V$,
		the $n+1$ vectors $v,Tv,\ldots, T^nv$ are linearly dependent.
		
		\item Using (1) and (2), deduce that when $k=\C$, there is a basis of $V$ in
		which the matrix of $T$ is upper-triangular.		
	\end{enumerate}
\end{problem}

\begin{problem}\label{F3Prob}
	Let $V:=\F_3[x]/(x^{12}),$ viewed as a vector space over the field $\F_3$ with $3$ elements. 
	Let $T:V\rightarrow V$ be the linear operator given by $(d/dx)^2$.
	\begin{enumerate}
		\item Find the matrix of $T$ with respect to the basis $\{x^i\}_{0\le i\le 11}$.
		\item Find bases for the kernel and image of $T$.
	\end{enumerate}
	

\end{problem}


\begin{problem}
	Let $T:V\rightarrow W$ be a linear transformation and $T^{\vee}:W^*\rightarrow V^*$
	the dual (transpose) map.  Prove that
	$\ker(T) = \im(T^{\vee})^{\perp}$ where for a subspace $U\subset V^*$ we set 
	$$U^{\perp}:=\{v\in V: u(v)=0\ \text{for all}\ u\in U\}$$
	Similarly, prove that $\im(T) = \ker(T^{\vee})^{\perp}$.
\end{problem}

\begin{problem}
	If $W\subseteq V$ is a subspace of a finite dimensional vector space with $\dim(W)=\dim(V)-1$,
	prove that there exists $f\in V^*$ with $W=\ker(f)$.
\end{problem}

\begin{problem}
	Suppose given a system of linear equations
	\begin{equation}
		\sum_{j=1}^n a_{ij} x_j = b_i\qquad i=1,\ldots, m.\label{sys}
	\end{equation}
	with $a_{ij},b_i\in \R$ for all $i,j$.  If (\ref{sys}) has a solution
	in $\C^n$, show that it has a solution in $\R^n$.  In this case, are all solutions
	in $\R^n$?
\end{problem}

\begin{problem}
	Using Gaussian elimination, show that any matrix over a field can be decomposed
	as a product $PLU$ with $P$ a square permutation matrix, $L$ a square lower-triangular 
	matrix, and $U$ a (non necessarily square) upper triangular matrix.
\end{problem}

\begin{problem}
	Let $k$ be a finite field of size $q$. 
	\begin{enumerate}
		\item Show that the number of $n\times n$ matrixes over $k$ is $q^{n^2}$.
		
		\item Find (and prove!) a formula for the number of invertible $n\times n$
		matrices over $k$.
				
		\item (Challenge) Find and prove a formula for the number of nilpotent
		$n\times n$ matrices over $k$.
		
	\end{enumerate}
\end{problem}





\section{Lecture 2}\label{lec2}


In what follows, unless stated to the contrary all vector spaces are finite dimensional over $\C$.
\begin{problem}
	Give necessary and sufficient conditions for an endomorphism $T:V\rightarrow V$ to have 
	a square root.
\end{problem}

\begin{problem}
	Prove that any $n\times n$ matrix over $\C$ is conjugate to its transpose.
\end{problem}

\begin{problem}
	Let $A$ and $B$ be $n\times n$ matrices.  Prove that $AB$ and $BA$
	have the same characteristic polynomial.  
\end{problem}


\begin{problem}
	The order $n$ Vandermonde determinant is
	\begin{equation*}
		V_n:=
		\begin{array}{|ccccc|}
			1 & x_1 & x_1^2 & \cdots & x_1^{n-1}\\
			1 & x_2 & x_2^2 & \cdots & x_2^{n-1}\\
			\vdots &\vdots & \vdots & \ddots & \vdots \\
			1 & x_n & x_n^2 & \cdots & x_n^{n-1}
		\end{array}
	\end{equation*}
	Prove that $(x_j-x_i)$ divides $V_n$ for all $i<j$ and conclude that
	$$V_n = \prod_{1\le i < j \le n} (x_j-x_i).$$
\end{problem}

\begin{problem}
	Prove that the set of diagonalizable $n\times n$ matrices in $\C^{n^2}$ is dense in the following sense:
	Given $A\in \Mat_{n\times n}(\C)$ and $\varepsilon > 0$, there exists $B\in \Mat_{n\times n}(\C)$
	all of whose entries are $< \varepsilon$ with $A+B$ diagonalizable.  Hint: first prove this for Jordan blocks.
\end{problem}

\begin{problem}
	If $T$ is an endomorphism of a vector space $V$ with $T^m=\id$, prove that $T$ is diagonalizable.
	Using your proof, find a more general criterion for diagonalizability.
\end{problem}

\begin{problem}
	Prove that a real $2\times 2$ matrix with positive off-diagonal entries is
	diagonalizable.
\end{problem}

\begin{problem}
	Give an example of a finite dimensional vector space $V$ over $\F_p$ and an endomorphism
	$T$ satisfying $T^m=\id$ for some $m$ which is {\em not} diagonalizable.
\end{problem}

\begin{problem}
	Let $T:V\rightarrow V$ be the linear map of problem \ref{F3Prob}.
	Determine the Jordan canonical form of $T$.
\end{problem}



\begin{problem}
	Let $A$ and $B$ be $n\times n$ matrices over a field $k$, and assume that 
	$AB-BA = B$.
	\begin{enumerate}
		\item If the characteristic of $k$ is zero or is larger than $n$, 
		prove that $B$ is nilpotent.
		
		\item If $\Char(k)=p$ and $p\le n$, is $B$ necessarily nilpotent?
	\end{enumerate}
\end{problem}

\begin{problem}
	Let $V:=\Mat_{n\times n}(\C)$ and for
	 a fixed $n\times n$ intertible matrix $A$, denote by $T_A:V\rightarrow V$
	 the linear map determined by $T_A(B)=ABA^{-1}$.
	 \begin{enumerate}
	 	\item If $A$ is diagonalizable, prove that $T_A$ is too.
		({\bf Hint:} Reduce to the case of diagonal $A$.)
		\item Is $T_A$ diagonalizable for general invertible $A$?
	 \end{enumerate}
	
	
\end{problem}


\begin{problem}
	For $T\in \Hom(V,V)$ and $v\in V$ the {\em cyclic subspace} $C(v)$ generated by $v$ is $\Span(\{T^i(v)\}_{i\ge 0})$.  
	\begin{enumerate}
		\item Show that $\dim(C(v))=1$ if and only if $v$ is an eigenvector.  
		
		\item More generally, relate $\dim(C(v))$ (for various $v$) to the Jordan form of $T$.
	\end{enumerate}
	
\end{problem}

\begin{problem}[Uniqueness of abstract Jordan form]
	Let $T\in \Hom(V,V)$ have (abstract) Jordan decomposition $T=S+N=S'+N'$
	with $S,S'$ semisimple and $N,N'$ nilpotent.
	\begin{enumerate}
		\item Prove that $S$ and $N$ can be taken to be polynomials in $T$. 
		\item Show that $SS'=S'S$ and $NN'=N'N$.
		\item Show that $S-S'$ is semisimple and $N-N'$ is nilpotent.
		\item Conclude that $S=S'$ and $N=N'$.
	\end{enumerate}
\end{problem}

\begin{problem}
	Determine the set of matrices which commute with a given matrix in each of the following
	cases:
	\begin{enumerate}
		\item $A$ is diagonal with distinct eigenvalues.
		\item $A$ is diagonal with not necessarily distinct eigenvalues.
		\item $A$ is a Jordan block.
		\item $A$ is in Jordan canonical form.
		\item General $A$.
	\end{enumerate}
\end{problem}

\begin{problem}
	For $n\ge 0$, let $F_n$ be the $n$-th Fibonacci number, defined recursively by $F_{0}:=0$, $F_1:=1$,
	and $F_n:=F_{n-1}+F_{n-2}$.  Find an explicit formula for the $n$-th Fibonacci number.
\end{problem}




\section{Lecture 3}\label{lec3}

\begin{problem}
   Let $V$ be a finite dimensional vector space over a field $k$, and $f$ a nondegenerate,
   symmetric bilinear form on $V$.  Prove that $\dim(W) + \dim(W^{\perp})=\dim V$
   for any $k$-subspace $W$ of $V$.
\end{problem}


\begin{problem}
	Using the Gram-Schmidt algorithm, prove that every invertible complex matrix can be written
	as a product $QR$ with $Q$ unitary and $R$ upper-triangular.  State and prove the version of
	this theorem over the real numbers.
\end{problem}

\begin{problem}
	Prove that any invertible real matrix $A$ can be written as a product $QP$ 
	with $Q$ orthogonal and $P$ symmetric and positive definite (called the polar decomposition
	of $A$).  Hint: Consider $A^tA$, which is symmetric and positive definite, and take its square
	root.  State and prove the version of this theorem over $\C$.
\end{problem}

\begin{problem}
	Let $V$ be a vector space over a field $k$ and $f:V\times V\rightarrow k$ a bilinear form.  	
	Prove that $f$ induces a nondegenerate form
	$$f: V/\ker(L_f) \times V/\ker(R_f)\rightarrow k$$
	and conclude that $V/\ker(L_f) \simeq (V/\ker(R_f))^*$ and hence that $\im(L_f)\simeq \im (R_f)^*$.
\end{problem}

\begin{problem}
	The {\em trace} of an $n\times n$ matrix $A$, denoted $\Tr(A)$, is the sum of the diagonal entries of $A$.
	Equivalently, it is the coefficient of $x^{n-1}$ in $\Char_A(x)$.  Using the fact that $\Tr(AB)=\Tr(BA)$,
	show that the pairing on $\Mat_{n\times n}(\R)$ given by $\langle A,B\rangle:=\Tr(AB)$
	is a symmetric bilinear form.  Determine its signature.
\end{problem}

\begin{problem}
	The following statement is false: if $T:V\rightarrow V$ is an endomorphism with $V$
	of finite dimension over $\C$, then there exists a positive definite Hermitian 
	form on $V$ with respect to which $T$ is normal.  
	\begin{enumerate}
		\item Explain why this statement is false.
		\item Add an hypothesis which makes the conclusion true.
		\item What happens if we change ``normal" to ``unitary" or to ``Hermitian"?
	\end{enumerate}
\end{problem}

\begin{problem}
	Let $P_n$ denote the real vector space of polynomials in one variable $x$ over $\R$ of degree $\le n$.
	Show that the pairing 
	$$\langle,\rangle: P_n\times P_n\rightarrow \R\qquad\text{given by}\qquad
	 \langle f,g\rangle:=\frac{d^n}{dx^n}(fg)\Bigg|_{x=0}$$
	 is a symmetric bilinear form, and compute its signature.
\end{problem}

\begin{problem}
	Let $P_2$ be the space of real polynomials in $x$ of degree $\le 2$, equipped with the
	inner product
	$$\langle f,g\rangle:= \int_{0}^1 fg\, dx.$$
	Find an orthonormal basis of $P_2$.
\end{problem}

\begin{problem}
	Prove that the set of unitary $n\times n$ matrices is a compact subset of $\C^{n^2}$.
\end{problem}

\begin{problem}
	Let $k$ be a field and $V$ a vector space over $k$. For a nondegenerate bilinear form $f:V\times V\rightarrow V$and a linear map $A:V\rightarrow V$, the {\em adjoint of $A$ with respect to $f$} is defined the composite
	\begin{equation*}
		\xymatrix{
			V \ar[r]^-{R_f} & V^* \ar[r]^-{A^{\vee}} & V^* \ar[r]^-{R_f^{-1}} \ar[r] & V
		}.
	\end{equation*}
	We denote the adjoint of $A$ with respect to $f$ by $A^*$ (and leave $f$ implicit).
	\begin{enumerate}
		\item Prove that $A^*$ is the unique linear map satisfying $f(Aw,v)=f(w, A^*v)$ for all $v$
		and $w$ in $V$.
		\item How does the notion of adjoint change if we  use $L_f$ in place of $R_f$ in the definition?
		\item Classify all nondegenerate pairings $f:V\times V\rightarrow V$ on the vector space $V:=k^n$
		with the property that for {\em all} linear maps $A:V\rightarrow V$,
		the matrix of $A^*$ in the standard basis of $V$ is the transpose of the matrix of $A$
		in the standard basis of $V$.
	\end{enumerate}
\end{problem}

\begin{problem}
	Suppose that $A$ is a real, symmetric matrix with the property that every negative eigenvalue of
	$A$ has even multiplicity.  Prove that $A$ has a real square root $B$.
\end{problem}

\begin{problem}
	Let $$A:=\begin{pmatrix}  \phantom{-}1 & \phantom{-}4 & -2 \\ \phantom{-}4 & \phantom{-}1 & -2 \\ -2 & -2 & -2 \end{pmatrix}.$$
	Find an orthogonal matrix $P$ so that $P^{-1}AP$ is diagonal.
\end{problem}


\begin{problem}
	Let $A$ be an $n\times n$ real skew-symmetric matrix.  
	\begin{enumerate}
		\item Prove that the nonzero eigenvalues of $A$ are purely imaginary.
		\item Prove that $\det(A + I_n) \ge 1$.
	\end{enumerate}
\end{problem}

\begin{problem}
	Let $V$ be the space of continuous real-valued and functions on the interval $[-\pi,\pi]$,
	equipped with the bilinear form
	$$\langle f, g\rangle :=\frac{1}{\pi}\int_{-\pi}^{\pi} f(x) g(x) dx$$
	and let $W$ be the subspace of $V$ spanned by the
	the collection of functions $B:=\{\frac{1}{\sqrt{2}}, \cos(kx), \sin(kx)\}_{k\ge 1}$.
	\begin{enumerate}
		\item Prove that $\langle \cdot,\cdot\rangle$ restricts to an inner product on $W$,
		and that $B$ is an orthonormal basis of $W$.
		\item Is $\langle \cdot, \cdot\rangle$ an inner product on $V$?  If so,
		can you describe the orthogonal complement of $W$?
	\end{enumerate}

\end{problem}





\bibliography{mybib}
\end{document}