\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}

\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}}

\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}
\newcommand{\ds}{\displaystyle}


\pagestyle{fancy} \lhead{{\sf Homework 5 $|$ Due February 19
    (Tuesday) }} \rhead{\thepage} \cfoot{{\sf MthSc 853 $|$ Linear
    Algebra $|$ Spring 2013 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Lax, Chapter 4, pages 32--43. \\


\begin{enumerate}
  
\item Let $T\colon X\to U$, with $\dim X=n$ and $\dim U=m$.  Show that
  there exist bases $B$ for $X$ and $B'$ for $U$ such that the matrix
  of $T$ in block form is
  \[
  M=\begin{bmatrix}I_k&0\\0&0\end{bmatrix}\,
  \]
  where $I_k$ is the $k\times k$ identity matrix, and the other blocks
  are either empty or contain all zeros. 



\item Consider the linear map $T:\R^3\to\R^3$ with matrix
  representation $\begin{bmatrix}1&-1&0 \\ 0&2&-2
    \\ -3&0&3\end{bmatrix}$ with respect to the standard basis. What
    is the matrix representation of $T$ with respect to the basis
    $\{(1,-1,0),(0,1,-1),(1,0,1)\}$?
    


\item Let $\mathcal{P}_n$ be the vector space of all polynomials over
  $\R$ of degree less than $n$.
  \begin{enumerate}
  \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. Indicate whether it is 1--1 or onto.
  \item Let $B_3=\{1,x,x^2\}$ be a basis for $\mathcal{P}_3$ and let
    $B_4=\{1,x,x^2,x^3\}$ be a basis for $\mathcal{P}_4$. Find the
    matrix representation of $T$ with respect to these bases.
  \end{enumerate}
  
  

\item Find necessary and sufficient conditions on the entries
  $u_1,u_2,u_3,u_4$ under which the following system of linear
  equations will have at least one solution over $\Z_5=\{0,1,2,3,4\}$,
  and give the number of solutions in case the conditions are met.
  \[
  \begin{bmatrix} 1&2&1&1 \\ 2&1&0&3 \\ 1&-1&-1&2 \\ 0&3&2&-1 \end{bmatrix}
  \vvvv{x_1}{x_2}{x_3}{x_4}=\vvvv{u_1}{u_2}{u_3}{u_4}.
  \]



\item Consider the system of linear equations $Tx=u$, where
  \[
  T=\begin{bmatrix} 0&2&-2 \\ 3&5&-2 \\ 4&0&4 \end{bmatrix},\qquad 
  u=\vvv{1}{-1}{1}.
  \]
  \begin{enumerate}
  \item Solve $Tx=u$ over $\R$.
  \item Solve $Tx=u$ over $\Z_2$.
  \end{enumerate}

\end{enumerate}

\end{document}







  
