\documentclass[12pt]{article}
\usepackage[sc]{mathpazo}
\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=-.1in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in
%%
\usepackage{fancyhdr,fancybox}
\usepackage{amssymb,amsmath,amsthm}
\usepackage{url}
\usepackage{hyperref}
\usepackage{latexsym}
\usepackage{enumitem} % Clashes with \usepackage{enumerate}
\usepackage{graphicx}
\graphicspath{{./}{figs/}{../figs}{../}}
\usepackage{tikz}
\usetikzlibrary{arrows.meta}

%% ------ Custom solution environment ---------
\usepackage{comment}
\specialcomment{solution}
  {\par\noindent\textbf{Solution.}\ } 
  {\par}                              
%\excludecomment{solution}  % COMMENT/UNCOMMENT this to show/hide solutions

%%----Custom Exercise environment-----
\newcounter{exer}    %% Make a special counter just for exercises
%\numberwithin{exer}{section}
\theoremstyle{definition}
\newtheorem{exmp}[exer]{Problem}
\newenvironment{exercise}[1][]
{\begin{exmp}[#1]}
{\end{exmp}}

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

\def\<{\langle}
\def\>{\rangle}
\def\longto{\longrightarrow}
\newcommand{\C}{\mathbb{C}} %% Some people prefer \CC or \CCC or \bbC
\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\def\And{\wedge}
\def\Or{\vee}
\def\Not{\neg}
%% \def\Not[1]{\overline{#1}}  %% Uncomment this for an alternative
\newcommand{\Red}[1]{\textcolor{red}{#1}}
\newcommand{\Blue}[1]{\textcolor{blue}{#1}}
\newcommand{\vv}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}}
\newcommand{\vvv}[3]{\begin{bmatrix} #1 \\ #2 \\ #3 \end{bmatrix}}
\newcommand{\vvvv}[4]{\begin{bmatrix} #1 \\ #2 \\ #3 \\ #4 \end{bmatrix}}
\newcommand{\ceil}[1] {\left\lceil #1 \right\rceil}
\newcommand{\floor}[1] {\left\lfloor #1 \right\rfloor}
\DeclareMathOperator{\Id}{Id}
\DeclareMathOperator{\lcm}{lcm}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\Trace}{tr}
\DeclareMathOperator{\Ker}{Ker}
\DeclareMathOperator{\Image}{Im}
\DeclareMathOperator{\tr}{tr}
\def\normal{\lhd}
\def\normaleq{\unlhd}
\def\nnormal{\ntriangleleft}
\def\nnormaleq{\ntrianglelefteq}

%% Sums, products, union & intersections, limits 
\def\Sum{\sum\limits}
\def\Prod{\prod\limits}
\def\Cap{\mathop\cap\limits}
\def\Cup{\mathop\cup\limits}

%% For vector and matrices
\def\A{\bold{A}}
\def\B{\bold{B}}
\def\I{\bold{I}}
\def\J{\bold{J}}
\def\P{\bold{P}}
\def\0{\bold{0}}
\def\a{\bold{a}}
\def\b{\bold{b}}
\def\c{\bold{c}}
\def\e{\bold{e}}
\def\p{\bold{p}}
\def\q{\bold{q}}
\def\t{\bold{t}}
\def\u{\bold{u}}
\def\v{\bold{v}}
\def\w{\bold{w}}
\def\x{\bold{x}}

%% Categories
\DeclareMathOperator\Hom{Hom}
\DeclareMathOperator\Ob{Ob}

%% Put these after \begin{solution}, as necessary
\newcommand\AIpolish{\textbf{(p)}\; }
\newcommand\AIminor{\textbf{(m)}\; }
\newcommand\AImajor{\textbf{(M)}\; }
\newcommand\AIpolishminor{\textbf{(p,m)}\; }
\newcommand\AIpolishmajor{\textbf{(p,M)}\; }

\renewcommand{\footrulewidth}{1pt}
\setlength{\headheight}{15pt}
\lhead{{\sf Summer Bridge Course (Algebra)}: Friday July 31}
\chead{}\rhead{\thepage}
\lfoot{Due Tuesdsay, August 4 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
\pagestyle{fancy}

\begin{document}

%\(\;\) \vspace{-4mm}

\noindent \textbf{Topics}: Singular value decomposition, pseudo-inverses. \medskip

\noindent \textbf{Do}: Answer the following questions. Assume that all matrices are over the field \(K=\mathbb{C}\). 

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

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

%% HW 18, Problem 1
\begin{exercise}
 Consider the following matrices
 \[
  A=\begin{bmatrix}4&4\\-3&3\end{bmatrix},\qquad
  A^T\!A=\begin{bmatrix}25&7\\7&25\end{bmatrix},\qquad
  AA^T=\begin{bmatrix}32&0\\0&18\end{bmatrix}.
  \]
  \begin{enumerate}[label=(\roman*)]
  \item Find the eigenvalues \(\sigma_1^2\), \(\sigma_2^2\) and unit
    eigenvectors \(v_1\), \(v_2\) of \(A^TA\). 
  \item For the \(\sigma_i\neq 0\), compute \(u_i=Av_i/\sigma_i\) and
    verify that indeed \(||u_i||=1\). Find the other \(u_i\) by computing
    the other unit eigenvector of \(AA^T\).
  \item Construct the singular value decomposition (SVD),
    \(A=U\Sigma V^T\).
  \item Write down orthonormal bases for each the ``four fundamental
    subspaces'': the column space \(R_A\), the nullspace \(N_A\), the row
    space \(R_{A^T}\), and the left nullspace \(N_{A^T}\).
  \item Describe \emph{all} matrices that have the same four fundamental
    subspaces.
  \item Find a left, right, and pseudoinverse of \(A\), or explain why
    it doesn't exist. 
  \end{enumerate}
\end{exercise}

%\begin{solution}
%\end{solution}

  %%------------------------------------------------------------------------
  
  %% HW 18, Problem 2
  \begin{exercise}
    Consider the matrix
    \(
    A=\begin{bmatrix} 1 & 0 & 1 \\ -1 & 1 & 0 \end{bmatrix}.
    \)
    \begin{enumerate}[label=(\roman*)]
    \item Construct the singular value decomposition of \(A\).
    \item Write down orthonormal bases for each the ``four fundamental
      subspaces'': the row space \(R_A\), the nullspace \(N_A\), the column
      space \(R_{A^T}\), and the left nullspace \(N_{A^T}\).
    \item Find a left inverse, right inverse, and pseudoinverse of \(A\),
      or explain why it doesn't exist. 
    \end{enumerate}
  \end{exercise}
  
%%------------------------------------------------------------------------

%% HW 18, Problem 3
\begin{exercise}
  Let \(f\colon\R^4\to\R^3\) be the linear map defined by \(f(x) =
  Mx\) for \(x\in\R^4\) where \(M=ABC\) and
  \[
  A = \begin{bmatrix} 1  & 1 & 1 \cr 1 & 1 & -1 \cr 1 &-2&0  \cr  \end{bmatrix}, \quad 
  B =  \begin{bmatrix}\frac{-1}{\sqrt{3}} & 0 & 0 & 0  \cr 0 & \frac{1}{\sqrt{6}} & 0 &0 \cr  0 &0  & -\sqrt{2} & 0 \cr    \end{bmatrix}, \quad 
  C =  \begin{bmatrix}1 & 1 & 1 & 1 \cr 1 & 1 & -1 & -1 \cr  1 & -1&1  & -1 \cr   1 & -1& -1 & 1 \cr  \end{bmatrix}.
  \]
  \begin{enumerate}
  \item Define the adjoint map \(f^*\colon\R^3 \to \R^4\) (under the standard Euclidean inner product) and express it in terms of \(M\).
  \item Find a singular value decomposition (SVD) of \(M\).
    (\emph{Hint}: Observe that \(A^TA\)  and \(C^TC\) are diagonal.)
  \item Find all \(x\in\R^4\) with \(\|x\|=1\) so that \(\|Mx\|\) is maximized.  
  \item Describe the eigenvalues and  eigenvectors of \(M^TM\).
  \item Find the least square solution for \(Mx=b\)  with  \(\|x\|_2\) minimal where \(b=(1,1,1)^T\). (\emph{Hint}: Use the pseudo-inverse of \(M\).)
  \end{enumerate}
\end{exercise}

%\begin{solution}
%\end{solution}

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

\end{document}