\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}
\def\normal{\lhd}
\def\normaleq{\unlhd}
\def\nnormal{\ntriangleleft}
\def\nnormaleq{\ntrianglelefteq}

%% 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)}: Wednesday July 22}
\chead{}\rhead{\thepage}
\lfoot{Due Thursday, July 23 2026}
\cfoot{}\rfoot{Written by M.~Macauley}
\pagestyle{fancy}


\begin{document}

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


\noindent \textbf{Topics}: The matrix of a linear map; change-of-basis
matrices. \\

\noindent \textbf{Do}: Answer the following questions. \\

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

% HW 12, Problem 1
\begin{exercise}
Let \(\mathcal{P}_n\) be the vector space of all polynomials over
  \(\R\) of degree less than \(n\).
\begin{enumerate}[label=(\roman*)]
  \item Show that the map \(T\colon\mathcal{P}_3\to\mathcal{P}_4\) given
    by
    \[
    T(p(x))=6\int_1^x p(t)\,dt
    \]
    is linear. Determine whether it is 1--1 or onto.
  \item Let \(\mathcal{B}_3=\{1,x,x^2\}\) be a basis for \(\mathcal{P}_3\) and let
    \(\mathcal{B}_4=\{1,x,x^2,x^3\}\) be a basis for \(\mathcal{P}_4\). Find the
    matrix representation of \(T\) with respect to these bases. \\
    \end{enumerate}
\end{exercise}

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

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

% HW 12, Problem 2
\begin{exercise}
Let \(T\colon X\to U\), with \(\dim X=n\) and \(\dim U=m\).  Show how
  to construct bases \(\mathcal{B}_X\) for \(X\) and \(\mathcal{B}_U\) for \(U\)
  such that the matrix of \(T\) in block form is
  \[
  M=\begin{bmatrix}I_r&0\\0&0\end{bmatrix},
  \]
  where \(I_r\) is the \(r\times r\) identity matrix, and the other blocks
  are either empty or contain all zeros. Prove/justify all of your claims. \\
\end{exercise}

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

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

% HW 12, Problem 3
\begin{exercise}
Let \(X\) be a vector space with basis \(x_1,x_2,x_3\), and consider the
  linear map \(T:X\to X\) with matrix representation
  \(\begin{bmatrix}1&-1&0 \\ 0&2&-2 \\ -3&0&3\end{bmatrix}\) with
    respect to this basis. What is the matrix
    representation of \(T\) with respect to the basis \(x_1-x_2\),
    \(x_2-x_3\), \(x_1+x_3\).
\end{exercise}

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

%%--------------------------------------------------------------------- 
    
\end{document}
