\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig,amsmath,latexsym,bm}
\usepackage{enumerate}
\usepackage{tikz}
\usepackage[all]{xy}
\usetikzlibrary{arrows}
\usetikzlibrary{decorations.markings}

\tikzset{big arrow/.style={decoration={markings,mark=at position 0.6
      with {\arrow[scale=2]{>}}}, postaction={decorate}, shorten >=1pt}}

% 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\C{\bold{C}}
\def\I{\bold{I}}
\def\M{\mathcal{M}}
\def\P{\bold{P}}
\def\U{\bold{U}}
\def\a{\bold{a}}
\def\b{\bold{b}}
\def\c{\bold{c}}
\def\e{\bold{e}}
\def\p{\bold{p}}
\def\u{\bold{u}}
\def\v{\bold{v}}
\def\x{\bold{x}}
\def\y{\bold{y}}
\def\<{\langle}
\def\>{\rangle}
\DeclareMathOperator{\Span}{Span}
\DeclareMathOperator{\rank}{rank}
\newcommand{\vv}[2]{\begin{bmatrix} #1 \\ #2 \end{bmatrix}}
\newcommand{\vvv}[3]{\begin{bmatrix} #1 \\ #2 \\ #3 \end{bmatrix}}

\newcommand{\ds}{\displaystyle}

\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 8 $|$ Due October 19 (Friday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 311 $|$ Linear Algebra $|$ Fall 2012 
    $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Strang, Section 4.1, 4.2. \\

\noindent\emph{Suggested short conceptual exercises}: Strang, Section
4.1, \#1, 2, 4, 5, 8--10, 13, 15, 18,--21, 24--29. Section 4.2, \#13,
18, 21--28.


\begin{enumerate}

  % Strang: Section 4.1, #3 (modified)
\item Construct a nonzero matrix $\A$ with the required property or say why
  it is impossible:
  \begin{enumerate}
  \item The column space contains $\vvv{1}{2}{-3}$ and
    $\vvv{3}{-1}{2}$, and the nullspace contains $\vvv{1}{1}{1}$.
  \item The row space contains $\vvv{1}{2}{-3}$ and $\vvv{3}{-1}{2}$,
    and the nullspace contains $\vvv{1}{1}{1}$.
  \item $\A\x=\vvv{1}{1}{1}$ has a solution and $\A^T \vvv{1}{0}{0} =
    \vvv{0}{0}{0}$
  \item Every row is orthogonal to every column.
  \item The sum of the columns is the zero vector, and the sum of the
    rows is a vector with all $1$'s.
\end{enumerate}



  % Strang: Section 4.1, #6, modified
\item Consider the following system of equations $\A\x = \b$:
  \begin{align*}
    x+2y+2z &= b_1 \cr
    2x + 2y + 3z &= b_2 \cr
    3x + 4y + 5z &= b_3.
  \end{align*}
  \begin{enumerate}
  \item Find numbers $y_1$, $y_2$, $y_3$ to multiply the left-hand
    sides of the equations so they add to $0$. You have found a vector
    $\y$ in which subspace? Write $\y^T\b=0$ in terms of $b_1$, $b_2$,
    and $b_3$?
  \item Using orthogonality of subspaces, what must be the case about
    $\y$ and $\b = (b_1,b_2,b_3)$ for there to be a solution to $\A\x =
    \b$? Does this condition hold for $\b = (5,5,9)$?
  \item What happens when we left-multiply both sides of the equation
    $\A\x=(5,5,9)$ by $\y^T$, where $\y$ is from Part~(a)?
  \end{enumerate}


  
  % Strang: Section 4.1, #11 (modified)
\item For each matrix, accurately sketch the four fundamental
  subspaces on two $\R^2$ plots so that orthogonal pairs of subspaces
  are plotted together. This is the ``grid picture.''
  \[
  \A=\begin{bmatrix}1&2\\3&6\end{bmatrix}\,\qquad\qquad\qquad
  \B=\begin{bmatrix}1&0\\3&0\end{bmatrix}\,.
  \]


  
  % Strang: Section 4.1, #17-18 (modified)
\item For a set $S$, let $S^\perp$ denote the \emph{orthogonal
  complement} of $S$, i.e., the set of vectors orthogonal to all
  vectors in $S$. Note that even if $S$ is not a subspace, $S^\perp$
  is.
  \begin{enumerate}
  \item If $S$ is the subspace of $\R^3$ containing only the zero
    vector, what is $S^\perp$? (Find a basis.)
  \item If $S$ is spanned by $(1,1,1)$, what is $S^\perp$? (Find a basis.)
  \item If $S$ is spanned by $(1,1,1)$ and $(1,1,-1)$, what is a basis
    for $S^\perp$? (Find a basis.)
  \item Now, suppose $S$ is not a subspace, but rather just the set
    containing the two vectors $(1,1,1)$ and $(1,1,-1)$. What is
    $S^\perp$? It is the nullspace of what matrix?
  \item Suppose $S$ is a set of vectors (not necessarily a
    subspace). Describe as concisely as possible what subspace
    $(S^\perp)^\perp$ is. What is the relation between $S$ and
    $(S^\perp)^\perp$ when $S$ actually is a subspace.
  \end{enumerate}



  % Strang: Section 4.2, #1, 3 (modified)
\item Let $\b=\vvv{1}{2}{2}$ and $\a=\vvv{1}{1}{1}$.
  \begin{enumerate}
  \item Project $\b$ onto the line through $\a$. Check that $\e=\b-\p$
    is orthogonal to $\a$.
  \item Find the projection matrix $\ds\P=\frac{\a\a^T}{\a^T\a}$ onto
    the line through $\a$. Verify that $\P^2=\P$. Multiply $\P\b$ to
    compute the projection $\p$.
  \end{enumerate}



  % Strang: Section 4.2, #5, 6, 7
\item Let $\a_1=\vvv{-1}{2}{2}$, $\a_2 = \vvv{2}{2}{-1}$, $\a_3 =
  \vvv{2}{-1}{2}$, and $\b=\vvv{1}{0}{0}$.
  \begin{enumerate}
  \item Compute the projection matrices $\P_1$ and $\P_2$ onto the
    lines through $\a_1$ and $\a_2$. Multiply those matrices and
    explain geometrically why $\P_1\P_2$ is what it is.
  \item Project $\b = (1,0,0)$ onto the lines through $\a_1$ and
    $\a_2$ and also onto $\a_3$. Add up the three projections
    $\p_1+\p_2+\p_3$.
  \item Find the projection matrix $\P_3$ onto $\a_3$. Verify that
    $\P_1+\P_2+\P_3=\I$. This means that the basis $\a_1,\a_2,\a_3$
    is orthogonal! (Think about why.)
\end{enumerate}



% Strang: Section 4.2, #17 (modified)
\item Suppose $\P$ is a projection matrix onto the column space of
  $\A$. 
  \begin{enumerate}
  \item Show that the matrix $\I-\P$ is also a projection matrix by
    verifying that $(\I-\P)^T=\I-\P$ and $(\I-\P)^2=\I-\P$ both hold.
  \item What subspace does the matrix $\I-\P$ project onto?
    [\emph{Hint}: Note that $\b=\P\b+(\I-\P)\b$ holds for any vector
      $\b$!]
  \end{enumerate}



% Strang: Section 4.2, #19, 20 (modified)
  \item Consider the plane $\mathcal{P}$ in $\R^3$ given by
    $x-y-2z=0$.
  \begin{enumerate}
  \item Find a matrix whose columns are a basis for $\mathcal{P}$.
  \item Compute $\P=\A(\A^T\A)^{-1}\A^T$, which is the projection
    matrix onto $\mathcal{P}$.
  \item Find a vector $\e$ that is orthogonal to
    $\mathcal{P}$. Compute the projection matrix
    $\bold{Q}=\e\e^T/\e^T\e$ and $\I-\bold{Q}$. How are $\P$ and
    $\bold{Q}$ related?
  \end{enumerate}


\end{enumerate}

\end{document}
