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\pagestyle{fancy} \lhead{{\sf Homework 6 $|$ Due February 26
    (Tuesday) }} \rhead{\thepage} \cfoot{{\sf MthSc 853 $|$ Linear
    Algebra $|$ Spring 2013 $|$ M.~Macauley}}


\begin{document}

$\;$

\noindent\emph{Read}: Lax, Chapter 5, pages 44--57. \\


\begin{enumerate}

% Lax Exercise 5.2-3
\item Let $S_n$ denote the set of all permutations of $\{1,\dots,n\}$.
  \begin{enumerate}
  \item Prove that $\sgn(\pi_1\circ\pi_2)=\sgn(\pi_1)\sgn(\pi_2)$.
%  \item Prove that if $\tau\in S_n$ is a transposition, then
%    $\sgn(\tau)=-1$.
  \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 $2$.
  \end{enumerate}



% Lax Exercise 5.5 with Halmos Exercises 31.2 and 31.3
\item Let $X$ be an $n$-dimensional vector space over a field $K$.
  \begin{enumerate}
  \item Prove that if the characteristic of $K$ is not $2$, then every
    skew-symmetric form is alternating.
  \item Give an example of a non-alternating skew-symmetric form.
  \item Give an example of a non-zero alternating $k$-linear form
    ($k<n$) such that $f(x_1,\dots,x_k)=0$ for some set of linearly
    independent vectors $x_1,\dots,x_k$.
  \end{enumerate}



\item Let $X$ be a $2$-dimensional vector space over $\C$, and let
  $f\colon X\times X\rightarrow\C$ be an alternating, bilinear
  form. If $\{x_1,x_2\}$ is a basis of $X$, determine a formula for
  $f(u,v)$ in terms of $f(x_1,x_2)$, and the coefficients used to
  express $u$ and $v$ with this basis. 
  


\item Let $X$ be an $n$-dimensional vector space over $\R$, and let
  $f$ be a non-degenerate symmetric bilinear form. That is, it has the
  additional property that for all nonzero $x\in X$, there is some
  $y\in X$ for which $f(x,y)\neq 0$.
  \begin{enumerate}
  \item Prove that if $f$ is non-degenerate, the map $L\colon
    X\rightarrow X'$ given by $L\colon x\mapsto f(x,-)$ is an
    isomorphism.
  \item Show that, given any basis $x_1,\dots,x_n$ for $X$, there
    exists a basis $y_1,\dots,y_n$ such that $f(x_i,y_j)=\delta_{ij}$.
  \item Conversely, prove that if $B_X=\{x_1,\dots,x_n\}$ and
    $B_Y=\{y_1,\dots,y_n\}$ are sets of vectors in $X$ with
    $f(x_i,y_j)=\delta_{ij}$, then $B_X$ and $B_Y$ are bases for $X$.
  \end{enumerate} 
    


\item Let $X$ be an $n$-dimensional vector space over $\R$, and let
  $f$ be a \emph{non-degenerate} symmetric bilinear form.
  \begin{enumerate}
  \item Show that there exists $x_1\in X$ with $f(x_1,x_1)\neq 0$.
  \item Fix $x_1\in X$ with $f(x_1,x_1)\neq 0$, and define $T$ by
    $T\colon x\mapsto f(x,x_1)$. What is the rank of $T$?
  \item Let $Z=\ker T$. Show that the restriction of $f$ to $Z\times
    Z$ is again non-degenerate.
  \item Prove that $X$ has a basis $\{x_1,\dots,x_n\}$ such that
    $f(x_i,x_i)\neq 0$ for all $i$.
  \item Prove or disprove that $f(x_i,x_j)=0$ whenever $i\neq j$.
  \item Give an example of a vector space $X$ ($2\leq \dim X<\infty$)
    with basis $B$ and a non-degenerate symmetric bilinear form $f$
    for which $f(x,x)=0$ for all $x\in B$.
  \end{enumerate}


  
  % Stated as fact in Proof of Lemma 5.4 in Lax
\item Let $A=(c_1,\dots,c_n)$ be an $n\times n$ matrix ($c_i$ is a
  column vector), and let $B$ be the matrix obtained from $A$ by
  adding $k$ times the $i^{\rm th}$ column of $A$ to the $j^{\rm th}$
  column of $A$, for $i\neq j$. Prove that $\det A=\det B$.
  

\end{enumerate}

\end{document}







  
