\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{\BBB}{\mathcal{B}}

\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}

\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 13
    (Friday) }} \rhead{\thepage} \cfoot{{\sf Math 8530 $|$ Linear
    Algebra $|$ Spring 2026 $|$ M.~Macauley}}


\begin{document}

$\;$ \vspace{-2mm}


\begin{enumerate}

  %%------------------------------------------------------------------------
  
\item This exercise will establish that permutations have a well-defined parity.
  \begin{enumerate}
  \item Prove that $\sgn(\pi_1\circ\pi_2)=\sgn(\pi_1)\sgn(\pi_2)$ for
    all $\pi_1,\pi_2\in S_n$.
  \item Prove that $\sgn(\tau)=-1$  for all transpositions $\tau\in S_n$.
  \item Let $\pi\in S_n$, and suppose that $\pi=\tau_k\circ\cdots\circ\tau_1=\sigma_\ell\circ\cdots\circ\sigma_1$, where $\tau_i,\sigma_j\in S_n$ are transpositions. Prove that $k\equiv\ell$ mod $2$. \medskip
  \end{enumerate}
  
  %%------------------------------------------------------------------------
  
\item Let $f$ be a \emph{non-degenerate} symmetric bilinear form over an $n$-dimensional vector space $X$. That is, for all nonzero $x\in X$, there is some $y\in X$ for which $f(x,y)\neq 0$. Consequently, fixing any nonzero $x\in X$ defines a nonzero dual vector
  \[
  f(x,-)\in X',\qquad f(x,-)\colon y\longmapsto f(x,y). 
  \]
  \begin{enumerate}
  \item Prove that the map $L_f\colon X\rightarrow X'$ given by $L_f\colon x\mapsto f(x,-)$ is an isomorphism.
  \item Let $x_1,\dots,x_n$ be a basis for $X$. Express the dual basis $\ell_1,\dots,\ell_n$ in this form. That is, find $g$ for which $L_g\colon x_i\mapsto \ell_i$.
  \item Show how to construct another basis $y_1,\dots,y_n$ such that $f(x_i,y_j)=\delta_{ij}$.
  \item Conversely, prove that if $\BBB_X=\{x_1,\dots,x_n\}$ and $\BBB_Y=\{y_1,\dots,y_n\}$ are sets of vectors in $X$ with $f(x_i,y_j)=\delta_{ij}$, then $\BBB_X$ and $\BBB_Y$ are bases for $X$. \medskip
  \end{enumerate} 
  
  %%------------------------------------------------------------------------
  
\item Let $f$ be a non-degenerate symmetric bilinear form over an
  $n$-dimensional vector space, where $1+1\neq 0$. 
  \begin{enumerate}
  \item Show that there exists $x_1\in X$ with $f(x_1,x_1)\neq 0$.
  \item Let $Z_1$ be the nullspace of $f(x_1,-)$. Show that $f$
    restricted to $Z_1$ is non-degenerate.
  \item Construct a basis $\{z_1,\dots,z_n\}$ for $X$ that satisfies
    $f(z_i,z_j)=\delta_{ij}$. \medskip
  \end{enumerate}
  
  %%------------------------------------------------------------------------
  
\item Let $f$ be an alternating bilinear form over a vector space $X$ with basis
  $\{x_1,x_2\}$. Determine a formula for $f(u,v)$
    in terms of each $f(x_i,x_j)$ and the coefficients used to express
    $u$ and $v$ with this basis. [Pun intended!] \medskip

    %%------------------------------------------------------------------------

\item Let $f$ be a symmetric bilinear form over a vector space $X$
  with basis $\{x_1,x_2\}$, and suppose that $f(x,x)=0$ for all $x\in
  X$. Find a formula for $f(u,v)$ in terms of each $f(x_i,x_j)$ and
  the coefficients used to express $u$ and $v$ with this basis. \medskip
  
  %%------------------------------------------------------------------------

\item Prove that $\tr(AB)=\tr(BA)$ for all $m\times n$ matrices $A$ and
    $n\times m$ matrices $B$. \medskip

  %%------------------------------------------------------------------------

\item If $A$ is square, write down a formula for $\tr(AA^T)$ in
  terms of $a_{ij}$. \medskip
    
  %%------------------------------------------------------------------------

\end{enumerate}

\end{document}








  
