\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig,amsmath,latexsym,bm}

% Code to make an augmented matrix (Usage: \begin{bmatrix}[ccc|c])
\makeatletter
\renewcommand*\env@matrix[1][*\c@MaxMatrixCols c]{%
  \hskip -\arraycolsep
  \let\@ifnextchar\new@ifnextchar
  \array{#1}}
\makeatother

\topmargin=-.5in
\headsep=0.2in
\oddsidemargin=0in
\textwidth=6.7in
\textheight=9.2in
\footskip=.5in

% Change this line to \def\bold{\mathbf} for ``upright boldface.''
\def\bold{\bm}

\newcommand{\F}{\mathbb{F}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Z}{\mathbb{Z}}
\def\0{\bold{0}}
\def\A{\bold{A}}
\def\B{\bold{B}}
\def\L{\bold{L}}
\def\M{\mathcal{M}}
\def\R{\bold{R}}
\def\U{\bold{U}}
\def\b{\bold{b}}
\def\c{\bold{c}}
\def\d{\bold{d}}
\def\u{\bold{u}}
\def\v{\bold{v}}
\def\x{\bold{x}}
\def\y{\bold{y}}
\def\<{\langle}
\def\>{\rangle}
\DeclareMathOperator{\Span}{Span}

\newcommand{\vv}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}}
\newcommand{\vvv}[3]{\begin{bmatrix} #1 \\ #2 \\ #3 \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}


\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due February 21 (Friday) }}
\rhead{\thepage} \cfoot{{\sf Math 311 $|$ Linear Algebra $|$ Spring 2014 
    $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Strang, Section 3.5, 3.6. \\

\noindent\emph{Suggested short conceptual exercises}: Strang, Section
3.5, \#11, 12, 14, 15, 18, 19, 21, 22, 24. Section 3.6, \#1, 7, 12,
13, 15, 16, 24--26. \\

\noindent Throughout, ``the four subspaces'' associated with a matrix
$\A$ refer to the column space $C(\A)$, row space $C(\A^T)$, nullspace
$N(\A)$, and left nullspace $N(\A^T)$. \smallskip


\begin{enumerate}


  % Strang: Section 3.5, #28 (highly modified) and 30
\item Let $\M$ be the space of all $2\times 3$ matrices.
  \begin{enumerate}
    \item Find a basis for the subspace $S$ consisting of the matrices
      in $\M$ whose columns add to zero.
    \item Find a basis for the subspace $T$ consisting of the matrices
      in $\M$ whose rows add to zero.
    \item Find a basis for the subspace $S\cap T$ of $\M$. This is the
      set of matrices whose rows and columns add to zero.
    \item Describe the subspace $S+T$, which is the set of all sums of
      matrices in $S$ with matrices in $T$.
    \item Compute $\dim(S+T)$, and express it in terms of $\dim S$,
      $\dim T$, and $\dim(S\cap T)$.
    \item Find a basis of the subspace $U$ of $\M$ whose nullspace
      contains $(2,1,1)$.
  \end{enumerate}



  % Made up myself (part (d) is Strang 3.5 #35)
\item Let $\mathcal{P}_3$ be the set of polynomials of degree at most
  $3$, which is a vector space.
\begin{enumerate}
  \item Determine a basis and the dimension of $\mathcal{P}_3$. 
  \item Explain why the set of polynomials of degree exactly $3$ is
    not a vector space. 
  \item The set of polynomials satifying $p''(x)=0$ is a subspace of
    $\mathcal{P}_3$. Find a basis for it and its dimension.
  \item Find a basis for the subspace of $\mathcal{P}_3$ consisting of
    the polynomials with $p(1)=0$.
\end{enumerate}
  


  % Strang: Section 3.5, #31 (highly modified)
\item The set $\mathcal{C}^\infty$ of smooth (i.e., infinitely
  differentiable) functions is an infinite dimensional vector
  space. Let $D$ be the differential operator $d/dx$, which is a
  \emph{linear map} from $\mathcal{C}^\infty$ to itself. (Being
  \emph{linear} just means that $D(af+bg)=aD(f)+bD(g)$ holds for all
  constants $a$ and $b$, and functions $f$ and $g$.)
  \begin{enumerate}
  \item What is the \emph{nullspace} of $D$? Write down an explicit
    basis and its dimension.
  \item Find a particular function $y_p(x)$ such that $Dy_p=3$.
  \item Use Parts~(a) and (b) to find \emph{all} functions that
    satisfy $Dy=3$. 
  \end{enumerate}



  % Made up myself
\item Let $D$ be the differential operator $d/dx-1$, which is a
  \emph{linear map} from the space of $\mathcal{C}^\infty$ of smooth
  functions to itself. Explicitly, if $y(x)$ is a function, then
  \[
  D\colon y(x)\longmapsto \left(\frac{d}{dx}-1\right)y(x)=y'(x)-y(x)\,.
  \]
  \begin{enumerate}
    \item The nullspace of $D$ is one-dimensional (you may assume this
      -- it is a well-known fact that an $n^{\rm th}$ order linear
      differential operator has an $n$-dimensional nullspace). Write
      down an explicit basis for the nullspace of $D$. [\emph{Hint}:
        You should be able to do this by inspection using only basic
        differential calculus.]
    \item Find a particular function $y_p(x)$ such that $Dy_p=3$,
      i.e., \emph{any} function satifying the equation
      $y'(x)-y(x)=3$. [\emph{Hint}: Try a constant function,
        $y_p(x)=c$. What $c$ works?]
    \item Use Parts~(a) and (b) to find \emph{all} functions that
      satisfy $Dy=3$. [This is the \emph{general solution} of the
        \emph{differential equation} $y'=y+3$. It's a vector space!]
  \end{enumerate}
 


  % Strang: Section 3.6, #2, 6 (modified)
\item Find bases and dimensions of the four subspaces associated
  with the following matrices:
  \[
  \A=\begin{bmatrix}1&2&4\\2&4&8\end{bmatrix},\qquad\quad
  \B=\begin{bmatrix}1&2&4\\2&4&7\end{bmatrix},\qquad\quad
  \bold{C}=\begin{bmatrix}0&2&2&2\\0&0&0&0\\0&1&0&1\end{bmatrix},\qquad\quad
  \bold{D}=\vvv{1}{3}{5}.
  \]
  You should be able to do this entirely by inspection; that is,
  without using elimination.



  % Strang: Section 3.6, #3 (modified)
\item Consider the matrix $\A$, whose $\L\U$-factorization is given below.
  Find a basis for and compute the dimension of each of the four subspaces
  associated with $\A$.
  \[
  \A=\begin{bmatrix}0&1&2&3&4\\0&1&2&4&7\\0&0&0&1&3\end{bmatrix}
  =\begin{bmatrix}1&0&0\\1&1&0\\0&1&1\end{bmatrix}
  \begin{bmatrix}0&1&2&3&4\\0&0&0&1&3\\0&0&0&0&0\end{bmatrix}=\L\U.
  \]
  Additionally, find $E=L^{-1}$, and write as $U=EA$.


  
  % Strang: Section 3.6, #5 (modified).
\item Let $P$ be the plane spanned by $(1,1,1)$ and $(1,2,0)$. 
  \begin{enumerate}
  \item Find a matrix that has $P$ as its column space.
  \item Find a matrix that has $P$ as its row space.
  \item Find a matrix that has $P$ as its nullspace.
  \item Find a matrix that has $P$ as its left nullspace.
  \end{enumerate}



  % Strang: Section 3.6, #11 (modified)
\item Let $\A$ be an $m\times n$ matrix of rank $r$. Suppose there are
  $n$-dimensional vectors $\b$ for which $\A\x=\b$ has no solution. 
  \begin{enumerate}
  \item List all inequalities ($<$ or $\leq$) that must hold between
    $m$, $n$, and $r$.
  \item Explain why $\A^T\y=\0$ must have a non-zero solution.
  \end{enumerate}


\end{enumerate}

\end{document}
