\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{\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\J{\bold{J}}
\def\L{\bold{\Lambda}}
\def\M{\bold{M}}
\def\N{\bold{N}}
\def\P{\bold{P}}
\def\Q{\bold{Q}}
\def\S{\bold{S}}
\def\U{\bold{U}}
\def\V{\bold{V}}
\def\e{\bold{e}}
\def\u{\bold{u}}
\def\v{\bold{v}}
\def\w{\bold{w}}
\def\x{\bold{x}}
\def\y{\bold{y}}
\def\<{\langle}
\def\>{\rangle}
\def\hat{\widehat}
\def\bar{\overline}

\DeclareMathOperator{\tr}{tr}
\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}}


\pagestyle{fancy} \lhead{{\sf Homework 14 $|$ Due December 7 (Friday) }}
\rhead{\thepage} \cfoot{{\sf MthSc 311 $|$ Linear Algebra $|$ Fall 2012 
    $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Strang, Section 6.7, 7.1, 7.2, 7.3. \\

\noindent\emph{Suggested short conceptual exercises}: Strang, Section
6.7, \#3, 9--13, 15. Section 7.1, \#1--12, 20--23, 29, 31. Section
7.2, \#10, 13, 17, 26--29, 32.



\begin{enumerate}

% Strang Section 6.7, #1-2 (modified)
\item Consider the following matrices:
  $\A=\begin{bmatrix}1&3\\2&6\end{bmatrix},\quad
  \A^T\A=\begin{bmatrix}5&15\\15&45\end{bmatrix},\quad
  \A\A^T=\begin{bmatrix}10&20\\20&40\end{bmatrix}$.
\begin{enumerate}
  \item Find the eigenvalues $\sigma_1^2$, $\sigma_2^2$ and unit
    eigenvectors $\v_1$, $\v_2$ of $\A^T\A$. 
  \item For the $\sigma_i\neq 0$, compute $\u_i=\A\v_i/\sigma_i$ and
    verify that indeed $||\u_i||=1$. Find the other $\u_i$ by computing
    the other unit eigenvector of $\A\A^T$.
  \item Write out the singular value decomposition (SVD),
    $\A=\U\bold{\Sigma}\V^T$.
  \item Write down orthonormal bases for the four fundamental
    subspaces of $\A$.
  \item Describe \emph{all} matrices that have the same four fundamental
    subspaces.
\end{enumerate}



% Strang Section 7.2, #3, 6 (modified)
\item Which of these transformations satisfy $T(\v+\w)=T(\v)+T(\w)$
  and which satisfy $T(c\v)=cT(\v)$? Assume that the mapping is either
  $\R^2\to\R^2$ or $\R^3\to\R^3$; it should be clear from the context
  which is which when it actually matters.
  \[
  \begin{array}{lllll}
    \text{(a)}\quad T(\v)=(v_2,v_1) && \text{(b)}\quad T(\v)=(v_1,v_1)
    && \text{(c)}\quad T(\v)=(0,v_1) \\ \text{(d)}\quad T(\v)=(0,1) &&
    \text{(e)}\quad T(\v)=v_1-v_2 && \text{(f)}\quad T(\v)=v_1v_2
    \\ \text{(g)}\quad T(\v)=\v/||\v|| && \text{(h)}\quad T(\v)=v_1+v_2+v_3 &&
    \text{(i)}\quad T(\v)=(v_1,2v_2,3v_3) \\
    \text(j)\quad T(\v)=\max\{v_i\}
  \end{array} 
  \]
  For those above that are indeed linear, write down the matrix $\A$
  of this transformation with respect to the standard unit basis
  vectors.

  

% Strang Section 7.2, #1 (highly modified)
\item Let $V$ be the space of all polynomials of degree at most
  $3$. Let $T\colon V\to V$ be the derivative operator $T=d/dx$ and
  $S\colon V\to V$ the second derivative operator $S=d^2/dx^2$. Use
  $1,x,x^2,x^3$ as both the input basis $\v_0,\v_1,\v_2,\v_3$ and the
  output basis $\w_0,\w_1,\w_2,\w_3$.
  \begin{enumerate}
  \item Write $T\v_0$, $T\v_1$, $T\v_2$, $T\v_3$ in terms of the
    $\w$'s and find the $4\times 4$ matrix $\A$ for $T$.
  \item Write $S\v_0$, $S\v_1$, $S\v_2$, $S\v_3$ in terms of the
    $\w$'s and find the $4\times 4$ matrix $\B$ for $T$.
  \item Compute the matrices $\A^2$, $\A\B$, $\B\A$, and $\B^2$. Which
    linear transformation (differential operator) do each of these
    products represent?
  \end{enumerate}
  


% Strang Section 7.2, #5-8 (modified) 
\item Let $T\colon V\to W$ be a linear transformation with $\dim V=\dim W=3$.
  \begin{enumerate}
  \item Using input basis $\v_1,\v_2,\v_3$ and output basis
    $\w_1,\w_2,\w_3$, suppose $T(\v_1)=\w_2$ and
    $T(\v_2)=T(\v_3)=\w_1+\w_3$. Find the matrix $\A$ for this
    transformation and multiply $\A$ by $(1,1,1)$. What is
    $T(\v_1+\v_2+\v_3)$?
  \item The \emph{kernel} of $T$, denoted $\ker T$, is defined as the
    set of all (input) vectors $\v$ for which $T(\v)=\0$. Find $\ker T$
    and the nullspace of $\A$. Find all solutions to $T(\v)=\w_2$.
  \item Find a vector $(c_1,c_2,c_3)$ that is not in the column space
    of $\A$. Find a combination of the $\w$'s that is not in the range
    of $T$.
\end{enumerate}



% Made up myself  
\item Give an explicit example of a linear transformation
  $T\colon\R^2\to\R^2$ with input basis $\v_1,\v_2$ and output basis
  $\w_1,\w_2$ such that the matrix for $T$ is $\A$ but the matrix for
  $T^2$ is \emph{not} $\A^2$. [\emph{Hint}: This will never work
  unless the $\v$'s are different from the $\w$'s.]



% Strang Section 7.2 #15-16 (modified)
\item
  \begin{enumerate}
    \item What matrix $\M$ transforms $(1,0)$ into $(r,t)$ and $(0,1)$
      into $(s,u)$?
    \item What matrix $\N$ transforms $(a,c)$ into $(1,0)$ and $(b,d)$
      into $(0,1)$?
    \item What condition on $a,b,c,d$ will make Part~(b) impossible?
    \item What matrix (in terms of $\M$ and $\N$) transforms $(a,c)$
      into $(r,t)$ and $(b,d)$ into $(s,u)$?
    \item What matrix transforms $(2,5)$ into $(1,1)$ and $(1,3)$ into
      $(0,2)$? Draw the ``grid picture'' of this transformation. 
  \end{enumerate} 



  % Made up myself
\item Let $T\colon V\to W$ be a linear transformation with $\dim V=n$
  and $\dim W=m$. Let $\v_1,\dots,\v_n$ be any basis for $V$. Describe
  precisely how to pick a basis $\w_1,\dots,\w_m$ for $W$ so that the
  $m\times n$ 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. The other three
  entries are blocks of zeros which are potentially empty (depending
  on the sizes $k$, $n$, and $m$). What does the number $k$ represent?



  % Made up myself
\item Consider the linear map $T:\R^3\rightarrow\R^3$ with matrix
  representation
  $\M=\begin{bmatrix}1&-1&0\\0&2&-2\\-3&0&3\end{bmatrix}$ with respect
  to the standard basis $\e_1$, $\e_2$, and $\e_3$.
    \begin{enumerate}
    \item What is the matrix representation $\A$ of $T$ with respect to the
      input basis $\v_1=(1,-1,0)$, $\v_2=(0,1,-1)$, $\v_3=(1,0,1)$?
      and standard output basis $\w_1=\e_1$, $\w_2=\e_2$, $w_3=\e_3$?
    \item What is the matrix representation $\B$ of $T$ with respect
      to the standard input basis $\v_1=\e_1$, $\v_2=\e_2$, $\v_3=\e_3$
      and the output basis $\w_1=(1,-1,0)$, $\w_2=(0,1,-1)$,
      $\w_3=(1,0,1)$?
    \item What is the matrix representation $\C$ of $T$ with respect
      to the basis $\v_1=\w_1=(1,-1,0)$, $\v_2=\w_2=(0,1,-1)$,
      $\v_3=\w_3=(1,0,1)$?
    \item For each of the three parts above, sketch a
      \emph{commutative diagram} relating the matrices $\M$ with (not
      necessarily all of) $\I$, $\A$, $\B$, and $\C$ and the matrix
      $\S$ whose columns are $(1,-1,0)$, $(0,1,-1)$, and $(1,0,1)$.
    \end{enumerate}


\end{enumerate}
  

\end{document}



