\documentclass[12pt]{article}
\usepackage{fancyhdr,fancybox,amssymb,epsfig, amsmath, latexsym, enumerate,
  mathrsfs}
\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}
\DeclareMathOperator{\sgn}{sgn}
\DeclareMathOperator{\tr}{tr}
\DeclareMathOperator{\Hom}{Hom}

\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 13 $|$ Due December 9
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Fall 2016 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Lax, Chapter 10, pages 143--153, and Appendix 4,
pages 313--316. \\

\noindent Let $U$ and $V$ be vector spaces over a field $K$. The \emph{tensor product} of $U$ and $V$ is the set $U\otimes V$ of finite sums of the form
\[
\sum c_i (u_i\otimes v_i)\,,
\]
where $c_i\in K$, $u_i\in U$, and $v_i\in V$, subject to the following relations:
\begin{align*}
  c(u\otimes v)&=(cu)\otimes v=u\otimes(cv)\,, \\
  (u+u')\otimes v&=u\otimes v+u'\otimes v\,, \\
  u\otimes(v+v')&=u\otimes v+u\otimes v'\,.
\end{align*}
Elements of the form $u\otimes v$ are called \emph{pure tensors}.



\begin{enumerate}

  \item Let $U$, $V$, and $X$ be vector spaces over a field
    $K$. Define a map
    \[
    \tau\colon U\times V\longrightarrow U\otimes V\,,\qquad
    \tau(u,v)=u\otimes v\,.
    \]
    \begin{enumerate}
    \item Prove that $\tau$ is bilinear.
    \item Prove that for any linear map $A\colon U\otimes V\to X$, the mapping
      $\alpha:=A\circ\tau$ is bilinear from $U\times V$ to $X$.
    \item Prove that for any bilinear map $\beta\colon U\times V\to
      X$, there is a unique linear mapping $A\colon U\otimes V\to X$
      such that $\beta=A\circ\tau$. This is the \emph{universal
        property} of the tensor product, and it says that any bilinear
      map can be ``factored through'' it. This is illustrated by the
      following commutative diagram:
      \[
      \xymatrix{ U\times V \ar[dr]_\tau\ar[rr]^\beta && X \\ &
        U\otimes V \ar@{-->}[ur]_A}
      \] \smallskip
    \end{enumerate}



  \item If $\{u_1,\dots,u_n\}$ and $\{v_1,\dots,v_m\}$ are bases for
    $U$ and $V$, respectively, then it is elementary to show that the
    pure tensors $\{u_i\otimes v_j\mid 1\leq i\leq n,\;1\leq j\leq
    m\}$ span $U\otimes V$. Show that these are linearly independent,
    and conclude that $\dim(U\otimes V)=(\dim U)(\dim
    V)$. [\emph{Hint}: Use the canonical basis $\{f_{ij}\}$ of
      the space of bilinear functions $U\times V\to K$, and use the
      universal property.] \\



  \item Use the universal property of the tensor product to prove the
    following results:
    \begin{enumerate}
    \item $U\otimes V\cong V\otimes U$ (hint: let $X=V\otimes U$);
    \item $(U\otimes V)\otimes W\cong U\otimes(V\otimes W)$;
    \item $(U\times V)\otimes W\cong (U\otimes W)\times (V\otimes W)$.
    \end{enumerate}

  
\end{enumerate}

\end{document}


%% CUT


% Lax Exercise A4.2 
\item Let $U$ and $V$ be vector spaces over a field $K$. Define the map 
  \[
  \varphi\colon U\otimes V\longrightarrow\Hom(U',V)\,,\qquad
  \varphi\colon u\otimes v\longmapsto \{\ell\mapsto(\ell,u)v\}\,,
  \]
  and extend linearily to arbitrary elements $\sum(u_i\otimes v_i)$ in
  $U\otimes V$. Prove that $\varphi$ is an isomorphism. \\


\item Define the \emph{index} of a real symmetric matrix $A$ to be the
  number of strictly positive eigenvalues minus the number of strictly
  negative eigenvalues. Suppose $A$ and $B$ are real symmetric
  matrices and $x^TAx\leq x^TBx$ for all $x\in X$. Prove that the
  index of $A$ is at most the index of $B$. \\
  
  
% Lax Exercise 10.3, 10.4, 10.5
\item Let $A$, $B$ be self-adjoint mappings with $0<A<B$. Here, $<$ is
  the partial ordering of self-adjoint mappings where $A<B$ iff $N-A$
  is positive-definite.
  \begin{enumerate}
  \item Show by example that the \emph{symmetrized product} $S:=AB+BA$
    need not be positive-definite.
  \item Show that $A^{1/k}<B^{1/k}$ if $k$ is a power of $2$.
  \item Show that $\log A\leq\log B$.
  \item Show by example that $A^2<B^2$ need not hold. \\
  \end{enumerate}


%\item Let $X_1$, $X_2$, $U_1$, and $U_2$ be vector spaces. Let
%  $A_1\in\Hom(X_1,U_1)$ and $A_2\in\Hom(X_2,U_2)$. Prove there is a
%  unique linear map $A_1\otimes A_2$ from $X_1\otimes X_2$ to
%  $U_1\otimes U_2$ satisfying $(A_1\otimes A_2)(x_1\otimes
%  x_2)=A_1(x_1)\otimes A_2(x_2)$.

